\(\Delta _{\underline{n}}f:=\Delta _{n_1}\cdots \Delta _{n_s}f=\prod _{\epsilon \in \lbrace 0,1\rbrace ^s}\mathcal {C}^{|\epsilon |}T^{\epsilon \cdot \underline{n}}f\)  (here we allow \(\underline{n}\in {\mathbb {Z}}^s\) ).
Given a system \((X,\mu ,T)\) 
we will use the seminorms \(\vert \!|\!| \cdot |\!|\!\vert _s\) , \(s\in {\mathbb {N}}\) , that were introduced in [1]} for ergodic systems and can be defined similarly for general systems (see for example [2]}). They are often refereed to as Gowers-Host-Kra seminorms and are inductively defined for \(f\in L^\infty (\mu )\)  as follows (for convenience we also define \(\vert \!|\!| \cdot |\!|\!\vert _0\) , which is not a seminorm):
\(\vert \!|\!| f|\!|\!\vert _0:=\int f\, d\mu ,\) 
and for \(s\in {\mathbb {Z}}_+\)  we let
\(\vert \!|\!| f|\!|\!\vert _{s+1}^{2^{s+1}}:=\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]} \vert \!|\!| \Delta _{n}f|\!|\!\vert _s^{2^{s}}.\) 
The limit can be shown to exist by successive applications of the mean ergodic theorem and for \(f\in L^\infty (\mu )\)  and \(s\in {\mathbb {Z}}_+\)  we have \(\vert \!|\!| f|\!|\!\vert _s\le \vert \!|\!| f|\!|\!\vert _{s+1}\)  (see [1]} or [2]}).
It follows immediately from the definition that
\(\vert \!|\!| f|\!|\!\vert _s^{2^s}=\lim _{N_1\rightarrow \infty }\cdots \lim _{N_s\rightarrow \infty }{\mathbb {E}}_{n_1\in [N_1]}\cdots {\mathbb {E}}_{n_s\in [N_s]} \int \Delta _{n_1,\ldots , n_s}f\, d\mu .\) 
It can be shown that we can take any \(s^{\prime }\le s\)  of the iterative limits to be simultaneous limits (i.e. average over \([N]^{s^{\prime }}\)  and let \(N\rightarrow \infty \) )
without changing the value of the limit. This was originally proved in [1]}, for a much simpler proof see [6]}.
For \(s^{\prime }=s\)  this gives the identity
\(\vert \!|\!| f|\!|\!\vert _s^{2^s}=\lim _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n}\in [N]^s} \int \Delta _{\underline{n}}f\, d\mu ,\) 
and for \(s^{\prime }=s-2\)  it gives the identity
\(\vert \!|\!| f|\!|\!\vert _s^{2^{s}}=\lim _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n}\in [N]^{s-2}} \vert \!|\!| \Delta _{\underline{n}}f|\!|\!\vert _2^{4}.\) 
Some key facts
Recall that if \((X,\mu ,T)\)  is a system, with \(\mathcal {E}(T)\)  we denote the set of its eigenfunctions with modulus one.
Proposition 3.1 
Let \((X,\mu ,T)\)  be an ergodic system and \(f\in L^\infty (\mu )\)  be a function with \(\left\Vert f\right\Vert _{L^\infty (\mu )}\le 1\) . Then
\(\vert \!|\!| f|\!|\!\vert _2^4\le \sup _{\chi \in \mathcal {E}(T)} \Re \Big ( \int f\cdot \chi \, d\mu \Big ).\) 

Let \(\mathcal {K}(T)\)  be the Kronecker factor of the system, meaning, the closed subspace of \(L^2(\mu )\)  spanned by all eigenfunctions of the system.
It is well known (and not hard to prove, see for example [2]}) that
\(\vert \!|\!| f|\!|\!\vert _{2}=\vert \!|\!| \tilde{f}|\!|\!\vert _2\) 
where \(\tilde{f}:={\mathbb {E}}(f|\mathcal {K}(T))\) .
Since the system is ergodic, the subspace \(\mathcal {K}(T)\)  has an orthonormal basis of eigenfunctions of modulus one, say \((\chi _j)_{j\in {\mathbb {N}}}\) . Then
\(\tilde{f}=\sum _{j=1}^\infty c_j\, \chi _j\)  where
\(c_j:=\int \tilde{f}\cdot \overline{\chi }_j\, d\mu =\int f\cdot \overline{\chi }_j\, d\mu ,\quad j\in {\mathbb {N}}.\) 
Then we have
\(\vert \!|\!| \tilde{f}|\!|\!\vert _2^4=\sum _{j=1}^\infty |c_j|^4\le \sup _{j\in {\mathbb {N}}}(|c_j|^2)\sum _{j=1}^\infty |c_j|^2=\sup _{j\in {\mathbb {N}}}(|c_j|^2)\left\Vert f\right\Vert _{L^2(\mu )}^2\le \sup _{j\in {\mathbb {N}}}\Big |\int f\cdot \overline{\chi }_j\, d\mu \Big |,\) 
where the first identity follows by orthonormality and direct computation, the second identity by the Parseval identity, and the last estimate since all functions involved are bounded by 1 .
The result now follows since the set \(\mathcal {E}(T)\)  is invariant under multiplication by unit modulus constants.
Proposition 3.2 
Let \((X,\mu ,T)\)  be an ergodic system and \(f\in L^\infty (\mu )\)  be such that \(\vert \!|\!| f|\!|\!\vert _{s+2}>0\)  for some \(s\in {\mathbb {Z}}_+\) . Then there exist \(\chi _{\underline{n}}\in \mathcal {E}(T)\) , \(\underline{n}\in {\mathbb {N}}\) , such that
\(\liminf _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in [N]^s}\Re \Big ( \int \Delta _{\underline{n}}f \cdot \chi _{\underline{n}}\, d\mu \Big )>0. \) 

By (REF ) we have that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n}\in [N]^s} \vert \!|\!| \Delta _{\underline{n}}f|\!|\!\vert _2^4>0.\) 
Using Proposition REF  we deduce that
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n}\in [N]^s}\sup _{\chi \in \mathcal {E}(T)} \Re \Big (\int \Delta _{\underline{n}}f\cdot \chi \, d\mu \Big )>0.\) 
From this the asserted estimate readily follows.
We will use the following variant of the so called Gowers-Cauchy-Schwarz inequality:
Lemma 3.3  Let \((X,\mu ,T)\)  be a system, and for \(s\in {\mathbb {N}}\)  let \(f_\epsilon \in L^\infty (\mu )\)  be bounded by 1 for \(\epsilon \in \lbrace 0,1\rbrace ^s\) , and \(g_{\underline{n}}\in L^\infty (\mu )\)  for \(\underline{n}\in {\mathbb {N}}^s\) . Let also \(\underline{1}:=(1,\ldots ,1)\) . Then for every \(N\in {\mathbb {N}}\)  we have
\(\Big |{\mathbb {E}}_{\underline{n}\in [N]^s}\, \int \prod _{\epsilon \in \lbrace 0,1\rbrace ^s} T^{\epsilon \cdot \underline{n}}f_\epsilon \cdot g_{\underline{n}}\, d\mu \Big |^{2^s}\le {\mathbb {E}}_{\underline{n},\underline{n}^{\prime }\in [N]^s}\, \int \Delta _{\underline{n}-\underline{n}^{\prime }}f_{\underline{1}}\cdot T^{-|\underline{n}|}\big (\prod _{\epsilon \in \lbrace 0,1\rbrace ^s}\mathcal {C}^{|\epsilon |}g_{\underline{n}^\epsilon }\big )\, d\mu .\) 

For notational simplicity we give the details only for \(s=2\) .
The general case follows in a similar manner by successively applying the Cauchy-Schwarz inequality with respect to the variables \(n_s,\ldots , n_1\) , exactly as we do below for \(s=2\) .
We have that
\(\Big |{\mathbb {E}}_{n_1,n_2\in [N]}\int f_0 \cdot T^{n_1}f_1 \cdot T^{n_2}f_2\cdot T^{n_1+n_2}f_3 \cdot g_{n_1,n_2}\, d\mu \Big |^2\) 
is bounded by (we use that \(f_0,f_1\)  are bounded by 1)
\({\mathbb {E}}_{n_1\in [N]}\int \Big |{\mathbb {E}}_{n_2\in [N]} T^{n_2}f_2\cdot T^{n_1+n_2}f_3 \cdot g_{n_1,n_2}\Big |^2\, d\mu .\) 
After expanding the square we find that this expression is equal to
\({\mathbb {E}}_{n_1\in [N]}\int {\mathbb {E}}_{n_2,n_2^{\prime }\in [N]}\, T^{n_2}f_2\cdot T^{n_2^{\prime }}\overline{f}_2\cdot T^{n_1+n_2}f_3\cdot T^{n_1+n_2^{\prime }}\overline{f}_3 \cdot g_{n_1,n_2}\cdot \overline{g_{n_1,n_2^{\prime }}} \, d\mu .\) 
After composing with \(T^{-n_2}\) , exchanging \({\mathbb {E}}_{n_1\in [N]}\)  with
\({\mathbb {E}}_{n_2,n_2^{\prime }\in [N]}\) ,
using the Cauchy-Schwarz inequality, and that \(f_2\)  is bounded by 1, we get that the square of the last expression is bounded by
\({\mathbb {E}}_{n_2,n_2^{\prime }\in [N]}\int \Big |{\mathbb {E}}_{n_1\in [N]} \, T^{n_1}f_3\cdot T^{n_1+n_2^{\prime }-n_2}\overline{f}_3 \cdot T^{-n_2}( g_{n_1,n_2}\cdot \overline{g}_{n_1,n_2^{\prime }}) \Big |^2 \, d\mu .\) 
As before, we expand the square, and compose with \(T^{-n_1}\) . We arrive at the expression
\({\mathbb {E}}_{n_1,n_2,n_1^{\prime },n_2^{\prime }\in [N]} \\\int f_3\cdot T^{n_1^{\prime }-n_1}\overline{f_3}\cdot T^{n_2^{\prime }-n_2}\overline{f_3} \cdot T^{n_1^{\prime }+n_2^{\prime }-n_1-n_2}f_3\cdot T^{-n_1-n_2}( g_{n_1,n_2}\cdot \overline{g}_{n_1,n_2^{\prime }}\cdot \overline{g}_{n_1^{\prime },n_2}\cdot g_{n_1^{\prime },n_2^{\prime }}) \, d\mu ,\) 
which is equal to the right hand side of the asserted estimate when \(s=2\) 
(for \(\underline{n}:=(n_1,n_2)\) , \(\underline{n}^{\prime }:=(n_1^{\prime },n_2^{\prime })\) ).
Combining the previous two estimates gives the asserted bound for \(s=2\) .
Lemma 3.4 
Let \((X,\mu , T)\)  be an ergodic system and \(f\in L^\infty (\mu )\)  be such that \(\vert \!|\!| f|\!|\!\vert _{s}=0\)  for some \(s\in {\mathbb {N}}\) .
For \(j=1,\ldots ,s\) , \(N\in {\mathbb {N}}\) , let \(b_{j,N}\in \ell ^\infty ({\mathbb {N}}^{s})\)  be sequences that do not depend on the variable \(n_j\) 
and are bounded by 1, and let
\(c_{N}(\underline{n}):=\prod _{j=1}^s b_{j,N}(\underline{n}), \quad \underline{n}\in [N]^{s}, \, N\in {\mathbb {N}}.\) 
Then
\(\lim _{N\rightarrow \infty } \left\Vert {\mathbb {E}}_{\underline{n}\in [N]^{s}}\, c_{N}(\underline{n})\cdot \Delta _{\underline{n}}f\right\Vert _{L^2(\mu )}=0.\) 

Using the Cauchy-Schwarz inequality, the fact that the sequence \(b_{s,N}\)  does not depend on the variable \(n_{s}\)  and is bounded by 1, and the identity
\( \Delta _{n_1,\ldots ,n_s}f=T^{n_s} (\Delta _{n_1,\ldots , n_{s-1}}f)\cdot \Delta _{n_1,\ldots , n_{s-1}}\overline{f},\) 
we get that it suffices to show that (for later convenience we rename the variable \(n_s\)  as \(n_s^{\prime }\) )
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in [N]^{s-1}}\left\Vert {\mathbb {E}}_{n_s^{\prime }\in [N]} \, \prod _{j=1}^{s-1} b_{j,N}(\underline{n}, n_s^{\prime })\cdot T^{n_s^{\prime }}(\Delta _{\underline{n}}f)\right\Vert ^2_{L^2(\mu )}=0.\) 
Using Lemma REF  for the average over \(n_s^{\prime }\)  and composing with \(T^{{-n_s^{\prime }}}\) , we get that it suffices to show that
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in [N]^{s-1}}{\mathbb {E}}_{n_s, n_s^{\prime }\in [N]} \, {\bf 1}_{[N]}(n_s+n_s^{\prime })\, \prod _{j=1}^{s-1}b_{j,N}(\underline{n}, n^{\prime }_s+n_s)\cdot \overline{b}_{j,N}(\underline{n}, n^{\prime }_s)\int T^{n_s}(\Delta _{\underline{n}}f)\cdot \Delta _{\underline{n}}\overline{f}\, d\mu =0,\) 
or, equivalently, that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n_s^{\prime }\in [N]} {\mathbb {E}}_{(\underline{n},n_s)\in [N]^s} \, {\bf 1}_{[N]}(n_s+n_s^{\prime }) \, \prod _{j=1}^{s-1}b_{j,N}(\underline{n}, n^{\prime }_s+n_s) \cdot \overline{b}_{j,N}(\underline{n}, n^{\prime }_s)\int \Delta _{\underline{n},n_s}f\, d\mu =0.\) 
This would follow if we show that
\(\lim _{N\rightarrow \infty } \sup _{n_s^{\prime }\in [N]}\Big | {\mathbb {E}}_{(\underline{n},n_s)\in [N]^s} \, {\bf 1}_{[N]}(n_s+n_s^{\prime }) \, \prod _{j=1}^{s-1}b_{j,N}(\underline{n}, n^{\prime }_s+n_s) \cdot \overline{b}_{j,N}(\underline{n}, n^{\prime }_s)\int \Delta _{\underline{n},n_s}f\, d\mu \Big | =0,\) 
or equivalently, that
for any choice of \(n^{\prime }_{s,N}\in [N]\) , \(N\in {\mathbb {N}}\) , we have
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{(\underline{n},n_s)\in [N]^s}\, {\bf 1}_{[N]}(n_s+n_{s,N}^{\prime }) \, \prod _{j=1}^{s-1}b_{j,N}(\underline{n}, n^{\prime }_{s,N}+n_s) \cdot \overline{b}_{j,N}(\underline{n}, n^{\prime }_{s,N})\int \Delta _{\underline{n},n_s}f\, d\mu =0.\) 
Using the Cauchy-Schwarz inequality we get that it suffices to show that
\(\lim _{N\rightarrow \infty } \left\Vert {\mathbb {E}}_{(\underline{n},n_s)\in [N]^s} \, \prod _{j=1}^{s-1}b^{\prime }_{j,N}(\underline{n},n_s)\cdot \Delta _{\underline{n},n_s}f\right\Vert _{L^2(\mu )}=0,\) 
where we let
\(b^{\prime }_{s-1,N}(\underline{n}, n_s):= {\bf 1}_{[N]}(n_s+n_{s,N}^{\prime }) \cdot b_{s-1,N}(\underline{n}, n^{\prime }_{s,N}+n_s)\cdot \overline{b}_{s-1,N}(\underline{n}, n^{\prime }_{s,N}) , \quad (\underline{n},n_s)\in [N]^s, \, N\in {\mathbb {N}},\) 
and for \(j=1,\ldots ,s-2\) , we let
\(b^{\prime }_{j,N}(\underline{n},n_s):= b_{j,N}(\underline{n}, n^{\prime }_{s,N}+n_s)\cdot \overline{b}_{j,N}(\underline{n}, n^{\prime }_{s,N}), \quad (\underline{n},n_s)\in [N]^s,\, N\in {\mathbb {N}}.\) 
Note that we now have a product of \(s-1\)  sequences, instead of \(s\) , and for \(j=1,\ldots , s-1\)  the sequences \(b^{\prime }_{j,N}\)  do not depend on the variable \(n_j\) . Hence, we can continue like that (on the next step we eliminate the sequences \(b^{\prime }_{s-1,N}\) , etc), and after \(s\)  steps we deduce that it suffices to show that
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in [N]^s} \,\int \Delta _{\underline{n}}f\, d\mu =0.\) 
This follows immediately from our assumption \(\vert \!|\!| f|\!|\!\vert _s=0\)  and (REF ) and completes the proof.

Joint ergodicity of general sequences
The primary goal of this
section is to prove Theorem REF . At the end of the section we also deduce Corollaries REF  and REF  from Theorem REF . For a better understanding of the argument,
we advice the reader to read first the technically less complicated model case that was treated in Section .
It is clear that Property \((i)\)  of Theorem REF  implies Property \((ii)\)  (use appropriate eigenfunctions to prove the equidistribution property).
So we only prove that Property \((ii)\)  implies Property \((i)\) .
Goal
It will be more convenient to prove the
following statement (the case \(l=\ell \)  implies Theorem REF ):
Proposition 4.1 
Let \((X,\mu ,T)\)  be an ergodic system and \(a_1,\ldots , a_\ell \colon {\mathbb {N}}\rightarrow {\mathbb {Z}}\) 
be sequences that are good for seminorm estimates and equidistribution for \((X,\mu ,T)\) . Then for every \(l\in \lbrace 0,1,\ldots , \ell \rbrace \) 
the following holds:
(\(P_l\) ) If \(f_1,\ldots ,f_\ell \in L^\infty (\mu )\)  are such that \(f_j\in \mathcal {E}(T)\)  for \(j=l+1,\ldots , \ell \) , then
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, T^{a_1(n)}f_1 \cdot \ldots \cdot T^{a_\ell (n)}f_\ell =\int f_1\, d\mu \cdot \ldots \cdot \int f_\ell \, d\mu \) 
where convergence takes place in \(L^2(\mu )\) .

In the following steps we fix \(\ell \in {\mathbb {N}}\) , a system, and a collection of sequences, and we are going to prove
that property \((P_l)\)  of Proposition REF  holds
by using (finite) induction on \(l\in \lbrace 0,1,\ldots , \ell \rbrace \) .

Proof strategy
Let us briefly describe the main idea of the proof, which is again motivated by the technique used in [8]}. If
(REF ) fails, then in Step 1 below, using our assumption that the sequences \(a_1,\ldots , a_\ell \)  are good for seminorm estimates for \((X,\mu ,T)\) , we deduce that \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+2}>0\)  for some \(s\in {\mathbb {Z}}_+\)  where \(\tilde{f}_l\)  is given by (REF ). This then implies that
\(\liminf _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in [N]^s}\,\Re \Big (\int \Delta _{\underline{n}}\tilde{f}_l\cdot \chi _{\underline{n}}\, d\mu \Big )>0,\) 
where \(\chi _{\underline{n}}\) , \(\underline{n}\in {\mathbb {N}}^s\) , are appropriate unit modulus eigenfunctions of the system \((X,\mu ,T)\) .
In Steps 2 and 3, using suitable manipulations of (REF ), the good equidistribution properties of the sequences \(a_1,\ldots , a_\ell \) , and the induction hypothesis, we deduce that the eigenfunctions \(\chi _{\underline{n}}\)  satisfy some algebraic relations, which allow us to conclude that they are products of sequences in \(s-1\)  variables.We crucially use in this step that the sequences \(a_1,\ldots , a_\ell \)  are good for equidistribution; if \(\ell \ge 3\)  and we only knew that they were good for irrational equidistribution (for example if they were polynomial sequences), then our argument would have been much more complicated (as is the case in [9]}). In view of this, (REF ) combined with Lemma REF  enables us to show that
\(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+1}>0\) . Iterating this step \(s+1\)  times we deduce that \(\int \tilde{f}_l\, d\mu >0\) . Given this, it is an easy matter in Step 4 to contradict the assumptions of Proposition REF .
Let us say a few words about the aforementioned “suitable manipulations” performed in Steps 2 and 3, since they are more complicated than those used for the case \(s=1\)  in Section .
Suppose for simplicity that \(\ell =l=s=2\) ;
hence \(\vert \!|\!| \tilde{f}_2|\!|\!\vert _{4}>0\) ,
and our goal is to show that \(\vert \!|\!| \tilde{f}_2|\!|\!\vert _{3}>0\) .
In this case, our assumption implies that
\(\liminf _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in [N]^2}\,\Re \Big (\int \Delta _{\underline{n}}\tilde{f}_2\cdot \chi _{\underline{n}}\, d\mu \Big )>0,\) 
for some eigenfunctions \(\chi _{\underline{n}}\)  with modulus 1. Using the Cauchy-Schwarz inequality twice (as in Lemma REF ) we deduce that (the symmetries of the ergodic seminorms are crucially used here)
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n},\underline{n}^{\prime }\in [N]^2}\,\limsup _{k\rightarrow \infty }{\mathbb {E}}_{n\in [N_k]}\, \Re \Big (\int \Delta _{\underline{n}-\underline{n}^{\prime }}g_k\cdot T^{a_1(n)}(\Delta _{\underline{n}-\underline{n}^{\prime }}\overline{f}_1) \cdot T^{a_2(n)}\chi _{\underline{n},\underline{n}^{\prime }} \, d\mu \Big )>0,\) 
where for \(\underline{n}=(n_1,n_2)\in [N]^2\) , \(\underline{n}^{\prime }=(n_1^{\prime },n_2^{\prime })\in [N]^2\) , the eigenfunctions \(\chi _{\underline{n},\underline{n}^{\prime }}\)  are given by
\(\chi _{\underline{n},\underline{n}^{\prime }}=\chi _{n_1,n_2}\cdot \overline{\chi }_{n_1,n_2^{\prime }}\cdot \overline{\chi }_{n_1^{\prime },n_2}\cdot \chi _{n_1^{\prime },n_2^{\prime }}.\) 
Note that the last estimate is substantially simpler to analyze than the \(\ell =2\)  case of (REF ), since in the inside average over the variable \(n\) , the iterate \(T^{a_2(n)}\)  is applied to an eigenfunction of the system and not to an arbitrary function in \(L^2(\mu )\) .
Using Property \((P_1)\)  of Proposition REF  (which is our induction hypothesis), we can assume that the function \(\Delta _{\underline{n}-\underline{n}^{\prime }}\overline{f}_1\)  is constant for all \(\underline{n},\underline{n}^{\prime }\in {\mathbb {N}}^2\) . Using the good equidistribution properties of the sequence \(a_2\) , we deduce that for \(N\in {\mathbb {N}}\)  there exist constants \(\underline{n}_N^{\prime }\in [N]^2\)  and sets \(\Lambda _N\subset [N]^2\)  with \(\liminf _{N\rightarrow \infty }|\Lambda _N|/N>0\) , such that \(\chi _{\underline{n},\underline{n}_N^{\prime }}\)  is constant for \(\underline{n}\in \Lambda _N\) .
Using (REF ), this gives that for those values of \(\underline{n}\in [N]^2\)  the 2-variable sequence \(\chi _{n_1,n_2}\)  is a product of two 1-variable sequences. If we use this information and an appropriate variant of (REF ), where we average over \(\Lambda _N\)  instead of \([N]^2\)  (by being more careful we can guarantee that this holds), we deduce by Lemma REF 
that \(\vert \!|\!| \tilde{f}_2|\!|\!\vert _{3}>0\) .

Step 1
(Characteristic factors)
The first step is to show that if (REF ) fails, then it also fails for some function \(f_l\)  of special form. The next statement can be proved
by adjusting the proof of Proposition REF  in a straightforward way, so we omit it.
Proposition 4.2 
Let \(a_1,\ldots , a_\ell \colon {\mathbb {N}}\rightarrow {\mathbb {Z}}\)  be sequences, \((X,\mu ,T)\)  be a system, and \(f_1,\ldots , f_\ell \in L^\infty (\mu )\)  be such that
\({\mathbb {E}}_{n\in [N]} \, T^{a_1(n)}f_1\cdots T^{a_\ell (n)}f_\ell \lnot \rightarrow 0\) 
in \(L^2(\mu )\)  as \(N\rightarrow \infty \) . Let \(l\in [\ell ]\) . Then there exist \(N_k\rightarrow \infty \)  and \(g_k\in L^\infty (\mu )\) , with \(\left\Vert g_k\right\Vert _{L^\infty (\mu )}\le 1\) , \(k\in {\mathbb {N}}\) , such that
for
\(\tilde{f}_l:=\lim _{k\rightarrow \infty } {\mathbb {E}}_{n\in [N_k]}\, T^{-a_l(n)}g_k\cdot \prod _{j\in [\ell ], j\ne l}T^{a_j(n)-a_l(n)}\overline{f}_j,\) 
where the limit is a weak limit (note that then \(\tilde{f}_1\in L^\infty (\mu )\) ), we have
\({\mathbb {E}}_{n\in [N]} \, T^{a_l(n)}\tilde{f}_l\cdot \prod _{j\in [\ell ], j\ne l}T^{a_j(n)}f_j \lnot \rightarrow 0\) 
in \(L^2(\mu )\)  as \(N\rightarrow \infty \) .


Step 2
(Seminorms of averages of functions)
Our next goal is to use Lemma REF  in order to show that if the seminorm of an average of functions is positive, then some related positiveness property holds for the individual functions
(the \(s=1\)  case was proved in Proposition REF ).
Proposition 4.3 
Let \((X,\mu ,T)\)  be an ergodic system, \(f_{n,k}\in L^\infty (\mu )\) , \(k,n\in {\mathbb {N}}\) , be bounded by 1, and \(f\in L^\infty (\mu )\)  be defined by
\(f:=\, \lim _{k\rightarrow \infty }{\mathbb {E}}_{n\in [N_k]}\, f_{n,k},\) 
for some \(N_k\rightarrow \infty \) ,
where the average is assumed to converge weakly.
If \(\vert \!|\!| f|\!|\!\vert _{s+2}>0\)  for some \(s\in {\mathbb {Z}}_+\) , then there exist
\(a>0\) , a subset \(\Lambda \)  of \({\mathbb {N}}^s\)  with positive lower density, and
\(\chi _{\underline{n}}\in \mathcal {E}(T)\) , \(\underline{n}\in \Lambda \) , such that
\(\Re \Big ( \int \Delta _{\underline{n}}f \cdot \chi _{\underline{n}}\, d\mu \Big )>a,\quad \underline{n} \in \Lambda ,\) 
and
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n},\underline{n}^{\prime }\in [N]^s}\,\limsup _{k\rightarrow \infty }{\mathbb {E}}_{n\in [N_k]}\, \Re \Big (\int \Delta _{\underline{n}-\underline{n}^{\prime }}f_{n,k} \cdot \chi _{\underline{n},\underline{n}^{\prime }}\cdot {\bf 1}_{\Lambda ^{\prime }}(\underline{n},\underline{n}^{\prime }) \, d\mu \Big )>0,\) 
where
\(\chi _{\underline{n},\underline{n}^{\prime }}:= T^{-|\underline{n}|}\big (\prod _{\epsilon \in \lbrace 0,1\rbrace ^s} \mathcal {C}^{|\epsilon |} \chi _{\underline{n}^\epsilon }\big ), \quad \underline{n},\underline{n}^{\prime }\in {\mathbb {N}}^s,\) 
and (recall that \(n^\epsilon \)  is defined in Section REF )
\(\Lambda ^{\prime }:=\lbrace (\underline{n},\underline{n}^{\prime })\in {\mathbb {N}}^{2s}\colon \underline{n}^\epsilon \in \Lambda \text{ for all } \epsilon \in \lbrace 0,1\rbrace ^s\rbrace .\) 

Remarks \(\bullet \)  The key point is that from a lower bound for expressions involving
\(\Delta _{\underline{n}}f\)  we infer lower bounds for expressions involving \(\Delta _{\underline{n}}f_{n,k}\) .
\(\bullet \)  For \(s=0\)  the conclusion is that there exists \(\chi \in \mathcal {E}(T)\)  such that \(\Re \big (\int f\cdot \chi \, d\mu \big )>0\) .
\(\bullet \) 
We plan to apply this proposition in the next step for the function \(\tilde{f}_l\)  given by (REF ) in place of \(f\) .

By Proposition REF  we have that there exist \(a>0\) , a subset \(\Lambda \)  of \({\mathbb {N}}^s\)  with positive lower density, and \(\chi _{\underline{n}}\in \mathcal {E}(T)\) , \(\underline{n}\in {\mathbb {N}}^s\) , such that
\(\Re \Big ( \int \Delta _{\underline{n}}f \cdot \chi _{\underline{n}}\, d\mu \Big )>a,\quad \underline{n} \in \Lambda .\) 
Hence,
\(\liminf _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in [N]^s}\, \Re \Big ( \int \Delta _{\underline{n}}f \cdot \chi _{\underline{n}}\cdot {\bf 1}_\Lambda (\underline{n})\, d\mu \Big )>0.\) 
Since \(\Delta _{\underline{n}}f=\prod _{\epsilon \in \lbrace 0,1\rbrace ^s}\mathcal {C}^{|\epsilon |}T^{\epsilon \cdot \underline{n}}f\) , \(\underline{n}\in {\mathbb {N}}^s\) , and
\(f=\lim _{k\rightarrow \infty }{\mathbb {E}}_{n\in [N_k]}\, f_{n,k}\)  (the limit is a weak limit), we deduce that
\(\liminf _{N\rightarrow \infty } \lim _{k\rightarrow \infty }{\mathbb {E}}_{n\in [N_k]}\, \Re \Big ({\mathbb {E}}_{\underline{n}\in [N]^s}\, \int \prod _{\epsilon \in \lbrace 0,1\rbrace ^k\setminus \lbrace \underline{1}\rbrace }\mathcal {C}^{|\epsilon |}T^{\epsilon \cdot \underline{n}}f \cdot T^{n_1+\cdots +n_s}f_{n,k}\cdot \chi _{\underline{n}}\cdot {\bf 1}_\Lambda (\underline{n})\, d\mu \Big )>0.\) 
For fixed \(k,n\in {\mathbb {N}}\)  we apply Lemma REF  for \(f_{\underline{1}}:=f_{n,k}\) , \(f_\epsilon :=\mathcal {C}^{|\epsilon |}f \)  for \(\epsilon \in \lbrace 0,1\rbrace ^s\setminus \underline{1}\) , and
\(g_{\underline{n}}:=\chi _{\underline{n}}\cdot {\bf 1}_\Lambda (\underline{n})\) , \(\underline{n}\in {\mathbb {N}}^s\) , and deduce that
\(\liminf _{N\rightarrow \infty } \limsup _{k\rightarrow \infty }{\mathbb {E}}_{n\in [N_k]}\,{\mathbb {E}}_{\underline{n},\underline{n}^{\prime }\in [N]^s}\, \int \Delta _{\underline{n}-\underline{n}^{\prime }}f_{n,k}\cdot T^{-|\underline{n}|}\Big (\prod _{\epsilon \in \lbrace 0,1\rbrace ^s}\big (\mathcal {C}^{|\epsilon |}\chi _{\underline{n}^\epsilon }\cdot {\bf 1}_\Lambda (\underline{n}^\epsilon )\big )\Big )\, d\mu >0.\) 
Since the limsup of a sum is at most the sum of the limsups, the second asserted estimate follows immediately from this one.
This completes the proof.

Step 3
(Going from \(\vert \!|\!| \cdot |\!|\!\vert _{s+2}\)  to \(\vert \!|\!| \cdot |\!|\!\vert _{1}\) )
Our next goal is to use Proposition REF  and the fact that the sequences \(a_1,\ldots , a_\ell \)  are good for equidistribution for the system \((X,\mu ,T)\)  in order to prove the following result:
Proposition 4.4 
Let \((X,\mu ,T)\)  be an ergodic system and
\(a_1,\ldots , a_\ell \colon {\mathbb {N}}\rightarrow {\mathbb {Z}}\)  be good for equidistribution for the system \((X,\mu ,T)\) . Suppose that property \((P_{l-1})\)  of Proposition REF  holds for some \(l\in [\ell ]\)  and let \(\tilde{f}_l\)  be as in (REF ), where all related functions are bounded by 1 and \(f_{l+1},\ldots , f_\ell \in \mathcal {E}(T)\) .
Suppose that \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+2}>0\)  for some \(s\in {\mathbb {Z}}_+\) . Then \(\int \tilde{f}_l\, d\mu \ne 0\) .

We can assume that the functions \(f_1,\ldots , f_\ell \)  are bounded by 1. We will show that if \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+2}>0\)  for some \(s\in {\mathbb {Z}}_+\) , then \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+1}>0\) . Applying this successively \(s+1\)  times, we deduce that \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{1}>0\) , or equivalently, that \(\int \tilde{f}_l\, d\mu \ne 0\) .
Since \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+2}>0\) , we can use Proposition REF  for \(\tilde{f}_l\)  in place of \(f\)  and
\(f_{n,k}:=T^{-a_l(n)}g_k\cdot \prod _{j\in [\ell ], j\ne l}T^{a_j(n)-a_l(n)}\overline{f}_j,\quad k,n\in {\mathbb {N}}.\) 
We deduce that there exist
\(a>0\) , a subset \(\Lambda \)  of \({\mathbb {N}}^s\)  with positive lower density,
\(\chi _{\underline{n}}\in \mathcal {E}(T)\) , \(\underline{n}\in {\mathbb {N}}^s\) , such that
\(\Re \Big ( \int \Delta _{\underline{n}}\tilde{f}_l \cdot \chi _{\underline{n}}\, d\mu \Big )>a, \quad \underline{n}\in \Lambda ,\) 
and
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n},\underline{n}^{\prime }\in [N]^s}\, \limsup _{k\rightarrow \infty }\\{\mathbb {E}}_{n\in [N_k]}\, \Re \Big (\int T^{-a_l(n)}(\Delta _{\underline{n}-\underline{n}^{\prime }}g_k) \cdot \prod _{j\in [\ell ], j\ne l}T^{a_j(n)-a_l(n)}(\Delta _{\underline{n}-\underline{n}^{\prime }}\overline{f}_j)\cdot \chi _{\underline{n},\underline{n}^{\prime }} \cdot {\bf 1}_{\Lambda ^{\prime }}(\underline{n},\underline{n}^{\prime })\, d\mu \Big )>0,\) 
where
\(\chi _{\underline{n},\underline{n}^{\prime }}:= T^{-|\underline{n}|}\big (\prod _{\epsilon \in \lbrace 0,1\rbrace ^s} \mathcal {C}^{|\epsilon |} \chi _{\underline{n}^\epsilon }\big ), \quad \underline{n},\underline{n}^{\prime }\in {\mathbb {N}}^d,\) 
and
\(\Lambda ^{\prime }:=\lbrace (\underline{n},\underline{n}^{\prime })\in {\mathbb {N}}^{2d}\colon \underline{n}^\epsilon \in \Lambda \text{ for all } \epsilon \in \lbrace 0,1\rbrace ^s\rbrace .\) 
Our goal is to use (REF ) in order to show that the eigenfunctions
\(\chi _{\underline{n}}\)  in (REF ) satisfy certain algebraic relations that enable us to
deduce, using Lemma REF , that \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+1}>0\) .
We thus start by analyzing (REF ). After composing with \(T^{a_l(n)}\) 
we get
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n},\underline{n}^{\prime }\in [N]^s}\, \limsup _{k\rightarrow \infty }\\{\mathbb {E}}_{n\in [N_k]}\, \Re \Big (\int \Delta _{\underline{n}-\underline{n}^{\prime }}g_k \cdot \prod _{j\in [\ell ], j\ne l}T^{a_j(n)}(\Delta _{\underline{n}-\underline{n}^{\prime }}\overline{f}_j)\cdot T^{a_l(n)} \chi _{\underline{n},\underline{n}^{\prime }} \cdot {\bf 1}_{\Lambda ^{\prime }}(\underline{n},\underline{n}^{\prime }) \, d\mu \Big )>0.\) 
We let
\(g_{j, \underline{n},\underline{n}^{\prime }}:=\Delta _{\underline{n}-\underline{n}^{\prime }}\overline{f}_j, \quad j\in [\ell ], \, j\ne l,\, \underline{n}, \underline{n}^{\prime }\in {\mathbb {N}}^s,\) 
and
\(g_{l, \underline{n},\underline{n}^{\prime }}:=\chi _{\underline{n},\underline{n}^{\prime }}, \quad \underline{n}, \underline{n}^{\prime }\in {\mathbb {N}}^s .\) 
Using the Cauchy-Schwarz inequality we deduce that
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n},\underline{n}^{\prime }\in [N]^s}\,{\bf 1}_{\Lambda ^{\prime }}(\underline{n},\underline{n}^{\prime }) \, \limsup _{k\rightarrow \infty }\left\Vert {\mathbb {E}}_{n\in [N_k]}\prod _{j=1}^\ell T^{a_j(n)}g_{j, \underline{n},\underline{n}^{\prime }}\right\Vert _{L^2(\mu )}>0.\) 
Since
\({\bf 1}_{\Lambda ^{\prime }}(\underline{n},\underline{n}^{\prime })=\prod _{\epsilon \in \lbrace 0,1\rbrace ^s}{\bf 1}_{\Lambda }(\underline{n}^\epsilon )\le {\bf 1}_{\Lambda }(\underline{n})\) 
and the set \(\Lambda \)  has positive lower density, we get
\(\liminf _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}^{\prime } \in [N]^s}{\mathbb {E}}_{\underline{n}\in \Lambda \cap [N]^s}\, \limsup _{k\rightarrow \infty }\left\Vert {\mathbb {E}}_{n\in [N_k]}\prod _{j=1}^\ell T^{a_j(n)}g_{j, \underline{n},\underline{n}^{\prime }}\right\Vert _{L^2(\mu )}>0.\) 
Note that since \(f_j\in \mathcal {E}(T)\)  for \(j=l+1,\ldots , \ell \) , we have
\(g_{j, \underline{n},\underline{n}^{\prime }}\in \mathcal {E}(T)\)  for \(j=l+1,\ldots , \ell \) , \(\underline{n},\underline{n}^{\prime }\in {\mathbb {N}}^s\) . Moreover,
since \(\chi _{\underline{n}}\in \mathcal {E}(T)\)  for all \(\underline{n}\in {\mathbb {N}}^s\)  we get that \(g_{l, \underline{n},\underline{n}^{\prime }}\in \mathcal {E}(T)\)  for all \(\underline{n},\underline{n}^{\prime }\in {\mathbb {N}}^s\) .
Since by assumption property \((P_{l-1})\)  of Proposition REF  holds, we get that the previous expression remains unchanged if for \(j=1,\ldots , l-1\)  we replace the functions
\(g_{j, \underline{n},\underline{n}^{\prime }}\) , \(\underline{n},\underline{n}^{\prime }\in {\mathbb {N}}^s\) , by constants (namely, their integrals).When we deal with polynomial sequences \(a_1,\ldots , a_\ell \) , we cannot replace
the functions \(g_{j, \underline{n},\underline{n}^{\prime }}\)  by constants but by their projection to the rational Kronecker factor, and this leads to serious complications. We bypass them in the proof of Theorem REF  by appealing to Theorem REF  below, which allows us to reduce matters to the case where the rational Kronecker factor is trivial.
Next, note that
\(T^{a_l(n)} \chi _{\underline{n}}=e(a_l(n)\alpha _{\underline{n}})\, \chi _{\underline{n}}, \quad n\in {\mathbb {N}},\, \underline{n}\in {\mathbb {N}}^s,\) 
for some \(\alpha _{\underline{n}}\in \operatorname{Spec}(T)\) , \(\underline{n}\in {\mathbb {N}}^s\) .
Hence,
we have that
\(T^{a_l(n)}g_{l,\underline{n},\underline{n}^{\prime }}=e(a_l(n)\beta _{\underline{n},\underline{n}^{\prime }})g_{l,\underline{n},\underline{n}^{\prime }},\) 
where
\(\beta _{\underline{n},\underline{n}^{\prime }}:=\sum _{\epsilon \in \lbrace 0,1\rbrace ^s} (-1)^{|\epsilon |}\alpha _{\underline{n}^\epsilon }.\) 
(For instance, if \(s=2\) , we have \( \beta _{\underline{n},\underline{n}^{\prime }}= \beta _{n_1,n_2,n_1^{\prime },n_2^{\prime }}:=\alpha _{n_1,n_2}-\alpha _{n_1^{\prime },n_2}-\alpha _{n_1,n_2^{\prime }}+\alpha _{n_1^{\prime },n_2^{\prime }}\) .)
For \(j=l+1,\ldots , \ell \) , \(\underline{n}, \underline{n}^{\prime }\in {\mathbb {N}}^s\) , the eigenvalues of the eigenfunctions \(g_{j,\underline{n},\underline{n}^{\prime }}\)  can be expressed in the form \(e(\alpha _{j,\underline{n},\underline{n}^{\prime }})\) , where \(\alpha _{j,\underline{n},\underline{n}^{\prime }}\in \operatorname{Spec}(T)\) . Then (recall the for \(j<l\)  the functions \(g_{j,\underline{n},\underline{n}^{\prime }}\)  are assumed to be constant)
\(\liminf _{N\rightarrow \infty }{\mathbb {E}}_{\underline{n}^{\prime } \in [N]^s}{\mathbb {E}}_{\underline{n}\in \Lambda \cap [N]^s}\, \limsup _{k\rightarrow \infty }| {\mathbb {E}}_{n\in [N_k]}\, e(a_l(n)\beta _{\underline{n},\underline{n}^{\prime }}+\sum _{j=l+1}^\ell a_j(n)\alpha _{j,\underline{n}, \underline{n}^{\prime }})|>0.\) 
From (REF ) we deduce that there exist integers \(\underline{n}^{\prime }_{N}\in [N]^s\) , and subsets \(\Lambda _{N}\)  of \(\Lambda \cap [N]^s\)  with
\(\liminf _{N\rightarrow \infty } \frac{|\Lambda _{N}|}{N^s}>0, \quad N\in {\mathbb {N}},\) 
and such that
\(\limsup _{k\rightarrow \infty } |{\mathbb {E}}_{n\in [N_k]}\, e(a_l(n)\beta _{\underline{n},\underline{n}^{\prime }_{N}}+\sum _{j=l+1}^\ell a_j(n)\alpha _{j,\underline{n}, \underline{n}^{\prime }_N})| >0 , \quad \underline{n}\in \Lambda _{N}.\) 
Since the sequences \(a_1,\ldots , a_\ell \)  are good for equidistribution for \((X,\mu ,T)\)  and
\(\lbrace \beta _{\underline{n},\underline{n}^{\prime }_{N}}\rbrace \) , \(\alpha _{j,\underline{n}, \underline{n}^{\prime }_N}\in \operatorname{Spec}(T) \) , \(j=l+1,\ldots , \ell \) , \(\underline{n}\in \Lambda _{N}, N\in {\mathbb {N}}\) ,
we deduce that
\(\beta _{\underline{n},\underline{n}_{N}^{\prime }}=0 \pmod {1}, \quad \underline{n}\in \Lambda _{N}, N\in {\mathbb {N}}.\) 
Then if \(\underline{n}_N^\epsilon :=(n_1^{\epsilon _1}, \ldots , n_s^{\epsilon _s})\) , where \(n_j^0:=n_j\)  and \(n_j^1:=n_{j,N}^{\prime }\)  for \(j=1,\ldots , s\) , we get
\(\alpha _{\underline{n}}=-\sum _{\epsilon \in \lbrace 0,1\rbrace ^s\setminus \underline{0}}(-1)^{|\epsilon |}\alpha _{\underline{n}_N^\epsilon } \pmod {1}, \quad \underline{n}\in \Lambda _{N},\, N\in {\mathbb {N}}.\) 
(For \(s=2\)  we get \(\alpha _{n_1,n_2}=\alpha _{n_{1,N}^{\prime },n_2}+\alpha _{n_1,n_{2,N}^{\prime }}-\alpha _{n_{1,N}^{\prime },n_{2,N}^{\prime }} \pmod {1}\) .)
The important point is that we expressed \(\alpha _{\underline{n}}\)  as a sum of sequences that depend on \(s-1\)  variables, chosen from the variables \(n_1,\ldots , n_s\) .
Since (REF ) holds for all \(\underline{n}\in \Lambda _{N}\subset \Lambda \) , \(N\in {\mathbb {N}}\) , and (REF ) holds, we deduce that
\(\liminf _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in \Lambda _{N}}\, \Re \Big ( \int \Delta _{\underline{n}}\tilde{f}_l \cdot \chi _{\underline{n}}\, d\mu \Big )>0.\) 
Finally we are going to combine this with (REF ) in order to deduce that \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+1}>0\) .
After
composing with the transformation \(T^{n_s^{\prime }}\) , averaging over \(n_s^{\prime }\in [N]\) , and using that
\(T^{n_s^{\prime }} \chi _{\underline{n}}=e(n_s^{\prime }\alpha _{\underline{n}})\cdot \chi _{\underline{n}},\quad \underline{n}\in {\mathbb {N}}^s,\, n_s^{\prime }\in {\mathbb {N}},\) 
and then using the Cauchy-Schwarz inequality,
we deduce that
\(\liminf _{N\rightarrow \infty } {\mathbb {E}}_{\underline{n}\in \Lambda _{N}}\, \left\Vert  {\mathbb {E}}_{n_s^{\prime }\in [N]} \,T_l^{n_s^{\prime }}(\Delta _{\underline{n}}\tilde{f}_l) \cdot e(n_s^{\prime } \alpha _{\underline{n}})\right\Vert _{L^2(\mu )}>0.\) 
Using Lemma REF  for the average over \(n_s^{\prime }\)  and composing with \(T^{-n_s^{\prime }}\)  on the integrals that arise, we deduce that
\(\liminf _{N\rightarrow \infty } \Re \Big ( {\mathbb {E}}_{(\underline{n},n_{s+1},n_s^{\prime })\in [N]^{s+2}}\, c_N(\underline{n}, n_{s+1},n_s^{\prime })\cdot {\bf 1}_{\Lambda _{N}}(\underline{n}) \int \Delta _{(\underline{n},n_{s+1})}\tilde{f}_l \, d\mu \Big )>0,\) 
where
\(c_N(\underline{n}, n_{s+1},n_s^{\prime }) := {\bf 1}_{[N]}(n_s^{\prime }+n_{s+1})\cdot e(n_{s+1} \alpha _{\underline{n}}), \quad \underline{n}\in [N]^s,\, n_{s+1}, n_s^{\prime }\in [N],\, N\in {\mathbb {N}}.\) 
Hence, for some \(n_{s,N}^{\prime }\in [N]\) , \(N\in {\mathbb {N}}\) , we have
\(\liminf _{N\rightarrow \infty } \Re \Big ( {\mathbb {E}}_{(\underline{n},n_{s+1})\in [N]^{s+2}}\, c_N(\underline{n}, n_{s+1},n_{s,N}^{\prime })\cdot {\bf 1}_{\Lambda _{N}}(\underline{n}) \int \Delta _{(\underline{n},n_{s+1})}\tilde{f}_l \, d\mu \Big )>0\) 
and using the Cauchy-Schwarz inequality we deduce that
\(\liminf _{N\rightarrow \infty } \left\Vert  {\mathbb {E}}_{(\underline{n},n_{s+1})\in [N]^{s+1}}\, c_N(\underline{n},n_{s+1},n_{s,N}^{\prime })\cdot {\bf 1}_{\Lambda _{N}}(\underline{n}) \cdot \Delta _{(\underline{n},n_{s+1})}\tilde{f}_l \right\Vert _{L^2(\mu )}>0.\) 
Using (REF ) and (REF ) we get that for all \(N\in {\mathbb {N}}\)  and \(\underline{n}\in \Lambda _{N}\)  we have
\(c_N(\underline{n}, n_{s+1}) =\prod _{j=1}^s b_{j,N}(\underline{n}, n_{s+1}), \qquad \underline{n}\in \Lambda _{N},\, n_{s+1}\in [N],\, N\in {\mathbb {N}},\) 
where \(b_{j,N}\)  do not depend on the variable \(n_j\) , for \(j=1,\ldots , s\) , \(N\in {\mathbb {N}}\) , and are bounded by 1. Since also \({\bf 1}_{\Lambda _{N}}(\underline{n})\)  does not depend on the variable \(n_{s+1}\) , we deduce
from Lemma REF  that
\(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+1}>0.\) 
This completes the proof of Proposition REF .

Step 4
(Proof of Proposition REF )
Let the ergodic system \((X,\mu ,T)\) , the positive integer \(\ell \) , and the sequences \(a_1,\ldots , a_\ell \)  be fixed.
We will prove that property \((P_l)\)  of Proposition REF  holds by induction on \(l\in \lbrace 0,\ldots , \ell \rbrace \) .
For \(\ell =1\) , as we mentioned in the introduction, it is a consequence of the spectral theorem that the asserted properties (\(P_0\)  and \(P_1\) ) hold. So we can assume that \(\ell \ge 2\) .
We first consider the case \(l=0\) . In this case our assumption is that
\(Tf_j=e(t_j)f_j\)  for some \(t_j\in \operatorname{Spec}(T)\) , \(j=1,\ldots , \ell \) . Note first that if \(t_j=0\)  for \(j=1,\ldots ,\ell \) , then by ergodicity we have that \(Tf_j=\int f_j \, d\mu \) , \(j=1,\ldots , \ell \) , in which case (REF ) is obvious. Hence, in order to show that (REF ) holds, it suffices to show that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, e(a_1(n)t_1+\cdots + a_\ell (n)t_\ell ) =0\) 
for \(t_1,\ldots , t_\ell \in \operatorname{Spec}(T)\) , not all of them zero. This holds since the sequences \(a_1,\ldots , a_\ell \)  are assumed to be good for equidistribution for the system \((X,\mu ,T)\) .
For \(l\in [\ell ]\)  we assume that property \((P_{l-1})\)  of Proposition REF  holds and we are going to show that property \((P_l)\)  holds.
Note that in order to prove that (REF ) holds it suffices to assume that at least one of the functions \(f_j\) , for \(j\in [\ell ]\setminus \lbrace l\rbrace \) , has zero integral. Indeed, if we write
\(f_j=\tilde{f}_j+\int f_j\, d\mu \)  where \(\tilde{f}_j=f_j-\int f_j\, d\mu \) , \(j\in [\ell ]\setminus \lbrace l\rbrace \) , and expand the product on the average into a sum of \(2^{\ell -1}\) 
terms, we get a sum of \(2^{\ell -1}-1\)  averages with functions that have the required property, plus the term \(\prod _{j\in [\ell ]\setminus \lbrace l\rbrace }\int f_j\, d\mu \cdot {\mathbb {E}}_{n\in [N]}T^{a_l(n)}f_l\) , which by the \(\ell =1\)  case we know that it converges in \(L^2(\mu )\)  to \(\prod _{j\in [\ell ]}\int f_j\, d\mu \) .
So under this additional assumption, our goal is to show that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, T^{a_1(n)}f_1 \cdot \ldots \cdot T^{a_\ell (n)}f_\ell =0\) 
where convergence takes place in \(L^2(\mu )\) . Arguing by contradiction, suppose that
\({\mathbb {E}}_{n\in [N]}\, T^{a_1(n)}f_1 \cdot \ldots \cdot T^{a_l(n)}f_\ell \lnot \rightarrow 0\) 
in \(L^2(\mu )\)  as \(N\rightarrow \infty \) .
Using Propositions REF  we get that the same thing holds with the function \(\tilde{f}_l\) ,
defined by (REF ), in place of \(f_l\) . Recall that
\(\tilde{f}_l=\lim _{k\rightarrow \infty } {\mathbb {E}}_{n\in [N_k]}\, T^{-a_l(n)}g_k\cdot \prod _{j\in [\ell ], j\ne l}T^{a_j(n)-a_l(n)}\overline{f}_j,\) 
for some \(N_k\rightarrow \infty \)  and some \(g_k\in L^\infty (\mu )\) , \(k\in {\mathbb {N}}\) , where all functions are bounded by 1, and the limit is a weak limit.
Since \(f_{l+1},\ldots , f_\ell \in \mathcal {E}(T)\) , we deduce that there exists a sequence of complex numbers \((c_n)\) , of modulus one, such that
\({\mathbb {E}}_{n\in [N]}\, c_n\, T^{a_1(n)}f_1 \cdot \ldots \cdot T^{a_l(n)}\tilde{f}_l \lnot \rightarrow 0\) 
in \(L^2(\mu )\)  as \(N\rightarrow \infty \) .
Using the fact that the sequences \(a_1,\ldots , a_\ell \)  are good for seminorm estimates for the system \((X,\mu ,T)\)  and the remark after the relevant definition in Section REF , we deduce that \(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s}>0\) , for some \(s\in {\mathbb {N}}\) .
Since the assumptions of Proposition REF  are satisfied, we deduce that \(\int \tilde{f}_l\, d\mu \ne 0\) .
Using (REF ), we deduce that
\(\lim _{k\rightarrow \infty } {\mathbb {E}}_{n\in [N_k]} \, \int T^{-a_l(n)}g_k\cdot \prod _{j\in [\ell ], j\ne l}T^{a_j(n)-a_l(n)}\overline{f}_j\, d\mu \ne 0.\) 
Composing with \(T^{a_l(n)}\)  and using the Cauchy-Schwarz inequality we get that
\({\mathbb {E}}_{n\in [N_k]} \, \prod _{j\in [\ell ], j\ne l}T^{a_j(n)}f_j\lnot \rightarrow 0\) 
in \(L^2(\mu )\)  as \(k\rightarrow \infty \) .
Since at least one of the functions \(f_j\) , for \(j\in [\ell ]\setminus \lbrace l\rbrace \) , has zero integral, and \( f_{l+1},\ldots , f_\ell \in \mathcal {E}(T)\) , using property \((P_{l-1})\)  of Proposition REF  (with \(f_l:=1\in \mathcal {E}(T)\) ) we get that the last limit is zero, a contradiction. We conclude that property \((P_l)\)  of Proposition REF  holds. This completes the induction and the proof of Proposition REF .

Proof of Corollary  REF
Let \(\mathcal {I}(T)\)  be the closed subspace of \(T\) -invariant functions. For \(f\in L^2(\mu )\) , we denote by \({\mathbb {E}}(f|\mathcal {I}(T))\)  the orthogonal projection, in \(L^2(\mu )\) , of \(f\)  onto \(\mathcal {I}(T)\) .
Using a standard argument we deduce from Theorem REF  that
the limit
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, \mu (A\cap T^{-a_1(n)}A\cap \cdots \cap T^{-a_\ell (n)}A)\) 
is equal to
\(\int ({\mathbb {E}}({\bf 1}_A|\mathcal {I}(T)))^{\ell +1}\, d\mu \ge \Big (\int {\mathbb {E}}({\bf 1}_A|\mathcal {I}(T))\, d\mu \Big )^{\ell +1}=(\mu (A))^{\ell +1}.\) 

Proof of Corollary  REF
It is clear that \((i)\)  implies \((ii)\) .
We show that \((ii)\)  implies \((i)\) .
Let \((X,\mu ,T)\)  be an ergodic \(k\) -step nilsystem for some \(k\in {\mathbb {N}}\) .
By [2]} we have that if \(f\in L^\infty (\mu )\)  is such that \(\vert \!|\!| f|\!|\!\vert _{k+1}=0\) ,
then \(f=0\) . Hence, any collection of sequences is good for seminorm estimates for \((X,\mu ,T)\) . Since, by assumption the
sequences \(a_1,\ldots , a_\ell \)  are good for equidistribution for \((X,\mu ,T)\) , it follows from Theorem REF  that they are jointly ergodic for \((X,\mu ,T)\) .

Characteristic factors of general sequences
In this section we prove Theorem REF  and use it to deduce Corollary REF .
Proof of Theorem  REF
In this subsection we show how we can deduce Theorem REF  from
Theorem REF .
For this, we will need some deeper tools from ergodic theory
than those need in the proof of Theorem REF . In [1]} it is shown that for every ergodic system \((X,\mathcal {X},\mu ,T)\)  and \(s\in {\mathbb {N}}\)  there exists
a \(T\) -invariant sub-\(\sigma \) -algebra \(\mathcal {Z}_s\)  of \(\mathcal {X}\)  with the property that for \(f\in L^\infty (\mu )\)  we have
\({\mathbb {E}}(f|\mathcal {Z}_s)=0\)  if and only if \(\vert \!|\!| f|\!|\!\vert _{s+1}=0\) .
We need some structural information about the
system \((X,\mathcal {Z}_s,\mu ,T)\)  that is a corollary of the main result in [1]}. We
remark that this is the only part of the article that we
make use of this structural theory.
Definition We say that a system \((X,\mu ,T)\)  has finite rational spectrum if \(\operatorname{Spec}(T)\cap \mathbb {Q}\)  is finite, or, equivalently, if there exists \(k\in {\mathbb {N}}\)  such that the ergodic components of the system \((X,\mu ,T^k)\)  have trivial rational spectrum.

Theorem 5.1 (Host-Kra [1]}) 
Let \((X,\mu ,T)\)  be an ergodic system such that \(\mathcal {X}=\mathcal {Z}_s\)  for some \(s\in {\mathbb {N}}\) . Then \((X,\mu ,T)\)  is an inverse limit of systems with finite rational spectrum.

Remark The main result in [1]} states much more, namely, \((X,\mu ,T)\)  is an inverse limit of nilsystems, but we will not need this.

We are also going to use the following fact:
Lemma 5.2 
Suppose that the sequences \(a_1,\ldots , a_\ell \colon {\mathbb {N}}\rightarrow {\mathbb {Z}}\)  are good for irrational equidistribution and seminorm estimates. For \(k\in {\mathbb {N}}\)  and \(i=1,\ldots , \ell \)  let
\(b_i(n):=\sum _{r=0}^{k-1} {\bf 1}_{k{\mathbb {Z}}+r}(a_i(n)) \, \frac{a_i(n)-r}{k}, \quad n\in {\mathbb {N}}.\) 
Then the sequences \(b_1,\ldots , b_\ell \)  are good for irrational equidistribution and seminorm estimates.

We first establish the equidistribution statement.
Let \(t_1,\ldots , t_\ell \in [0,1)\) , not all of them rational.
We want to show that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, e\big (\sum _{i=1}^\ell b_i(n)t_i\big )=0.\) 
By direct computation we see that it suffices to show that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, F(a_1(n),\ldots , a_\ell (n))\, e\big (\sum _{i=1}^\ell a_i(n)\frac{t_i}{k}\big )=0,\) 
where \(F\colon {\mathbb {Z}}_k^\ell \rightarrow  is defined by\)\(F(t_1,\ldots , t_\ell ):=e\big (-\sum _{i=1}^\ell \sum _{r=1}^k {\bf 1}_{k{\mathbb {Z}}+r}(x_i)\frac{r}{k}t_i\big ), \quad t_1,\ldots , t_\ell \in {\mathbb {Z}}_k.\)\(Using the Fourier expansion of \) F\( on \) Zk\( we see that it suffices to show that\)\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, e\big (\sum _{i=1}^\ell a_i(n)\frac{t_i+r_i}{k}\big )=0\)\(for all \) r1,..., r{0,..., k-1}\(. Since, by assumption, the sequences \) a1,..., a\( are good for irrational equidistribution, and \) ti+rik\( is irrational for some \) i{1,..., }\(, the needed identity follows.\)
Next, we establish the statement about seminorm estimates.
Let \((X,\mu ,T)\)  be a system and \(k\in {\mathbb {N}}\) . Our plan is to show that
the sequences \(b_1,\ldots , b_\ell \)  are good for seminorm estimates for the system \((X,\mu ,T)\)  by
using that (by our assumption) the sequences \(a_1,\ldots , a_\ell \)  are good for seminorm estimates for the “\(k\) -th root” of the system \((X,\mu ,T)\) , which is defined as follows:
We consider the system \((X_k,\mu _k,T_k)\)  where
\(X_k:=X\times \lbrace 0,\ldots , k-1\rbrace , \quad \mu _k=\mu \times \nu _k, \quad \nu _k:=\frac{\delta _0+\cdots +\delta _{k-1}}{k},\) 
and for \(x\in X\)  we let
\(T_k(x,i)=(x,i+1), \, i=0,\ldots , k-2, \quad T_k(x,k-1)=(Tx,0).\) 
The key property is that
\(T_k^k(x,i)=(Tx,i), x\in X, i\in \lbrace 0,\ldots , k-1\rbrace \) , hence for \(f\in L^\infty (\mu )\)  we have
\((T_k^k(f\otimes 1))(x,i)=(Tf)(x), \quad x\in X, \, i\in \lbrace 0,\ldots , k-1\rbrace .\) 
Applying our assumption for the system \((X_k,\mu _k,T_k)\) ,
and taking into account the remarks following the definition in Section REF  for the good seminorm property, we get that
there exists an \(s\in {\mathbb {N}}\)  (we can assume that \(s\ge 2\) )
such that if \(g_1, \ldots , g_\ell \in L^\infty (\mu _k)\)  and
\(\vert \!|\!| g_l|\!|\!\vert _{s,T_k}=0\)  for some \(l\in [\ell ]\) , then for every bounded sequence \((c_n)\)  we have
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, c_n\, T_k^{a_1(n)}g_1\cdot \ldots \cdot T_k^{a_l(n)}g_l= 0\) 
in \(L^2(\mu _k)\) .
We claim that this \(s\in {\mathbb {N}}\)  produces good seminorm estimates for the sequences \(b_1,\ldots , b_\ell \)  for the system \((X,\mu ,T)\) . To see this let \(f_1,\ldots , f_\ell \in L^\infty (\mu )\)  be such that \(\vert \!|\!| f_l|\!|\!\vert _{s,T}=0\)  for some \(l\in [\ell ]\) . It suffices to show that
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, T^{b_1(n)}f_1\cdot \ldots \cdot T^{b_l(n)}f_l= 0\) 
in \(L^2(\mu )\) .
Let \(g_i\in L^\infty (\mu _k)\)  be defined by
\(g_i:=f_i\otimes 1\) , \(i=1,\ldots , \ell \) , and recall that \(\mu _k=\mu \times \nu _k\) . By (REF ) it suffices to show that
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, T_k^{kb_1(n)}g_1\cdot \ldots \cdot T_k^{ka_l(n)}g_l= 0\) 
in \(L^2(\mu _k)\) , or equivalently, that
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, T_k^{a_1(n)}(T_k^{-r_1(n)}g_1)\cdot \ldots \cdot T_k^{a_l(n)}(T_k^{-r_l(n)}g_l)= 0\) 
in \(L^2(\mu _k)\) , for some sequences \(r_1,\ldots , r_l\colon {\mathbb {N}}\rightarrow \lbrace 0,\ldots , k-1\rbrace \) . Note that
the last average can be written as a sum of \(k^l\)  weighted averages of the form
\({\mathbb {E}}_{n\in [N]}\, c_n\, T_k^{a_1(n)}(T_k^{-r_1}g_1)\cdot \ldots \cdot T_k^{a_l(n)}(T_k^{-r_l}g_l)\) 
for some \(r_1,\ldots , r_l\in \lbrace 0,\ldots , k-1\rbrace \)  and \(c_n\in \lbrace 0,1\rbrace \) , \(n\in {\mathbb {N}}\) . Hence, it suffices to show that averages of the form (REF )
converge to 0 in \(L^2(\mu _k)\)  and \(N\rightarrow \infty \) .
Since \(g_l=f_l\otimes 1\)  and \(\mu _k=\mu \times \nu _k\) , our assumption \(\vert \!|\!| f_l|\!|\!\vert _{s,T}=0\)  and (REF ) give that
\(\vert \!|\!| g_l|\!|\!\vert _{s,T_k^k}=\vert \!|\!| f_l|\!|\!\vert _{s,T}=0\) .
Since \(s\ge 2\) , by [15]} we have that \(\vert \!|\!| g_l|\!|\!\vert _{s,T_k^k}=0\)  implies that \(\vert \!|\!| g_l|\!|\!\vert _{s,T_k}=0\) , hence we also have \(\vert \!|\!| T_k^{-r_l}g_l|\!|\!\vert _{s,T_k}=\vert \!|\!| g_l|\!|\!\vert _{s,T_k}=0\) . It then follows from (REF ) that the averages (REF ) converge to 0 in \(L^2(\mu _k)\)  as \(N\rightarrow \infty \) . This completes the proof.
It is straightforward to verify that Property \((i)\)  implies Property \((ii)\)  (the seminorm property holds with \(s=2\)  and one can use appropriate rotations on \(\ell \)  to verify the equidistribution property).
So we only prove that Property \((ii)\)  implies Property \((i)\) .
Let \(\mu =\int \mu _x\, d\mu \)  be the ergodic decomposition of the measure \(\mu \) . Since
\({\mathbb {E}}(f|\mathcal {K}_{rat}(\mu ))=0\)  implies that \({\mathbb {E}}(f|\mathcal {K}_{rat}(\mu _x))=0\)  for \(\mu _x\)  almost every \(x\in X\)  (see for example [16]}), we can assume that the system \((X,\mu ,T)\)  is ergodic.
We claim that it suffices to show that the Kronecker factor \(\mathcal {K}(T)\)  is characteristic for mean convergence of the averages
\({\mathbb {E}}_{n\in [N]}\, T^{a_1(n)}f_1 \cdot \ldots \cdot T^{a_\ell (n)}f_\ell ,\) 
meaning that the previous averages converge to 0 in \(L^2(\mu )\)  as \(N\rightarrow \infty \)  if at least one of the functions is orthogonal to \(\mathcal {K}(T)\)  (recall that \(\mathcal {K}(T)\)  is \(L^2(\mu )\) -closure of the linear span of the eigenfunctions of the system).
Indeed, if this is the case, then by approximation, we can assume that all functions are eigenfunctions; hence, for \(i=1,\ldots , \ell \)  we have
\(T^{a_i(n)}f_i=e(a_i(n)t_i)\, f_i, \quad n\in {\mathbb {N}},\) 
for some \(t_i\in [0,1)\) , not all of them rational (since at least one of the functions is orthogonal to \(\mathcal {K}_{rat}(T)\) ). In this case, the needed convergence to zero follows from
our assumption that the sequences \(a_1,\ldots , a_\ell \)  are good for irrational equidistribution.
So it remains to show that if \({\mathbb {E}}(f_i|\mathcal {K}(T))=0\) , for some \(i\in \lbrace 1,\ldots , \ell \rbrace \) , then the averages (REF ) converge to 0 in \(L^2(\mu )\) . Without loss of generality we can assume that \(i=1\) , hence \({\mathbb {E}}(f_1|\mathcal {K}(T))=0\) .
Since the sequences \(a_1,\ldots , a_\ell \)  are very good for seminorm estimates and \({\mathbb {E}}(f|\mathcal {Z}_s(T))=0\)  implies \(\vert \!|\!| f|\!|\!\vert _{s+1,T}=0\) , we can assume that all functions \(f_1,\ldots , f_\ell \)  are \(\mathcal {Z}_s(T)\) -measurable for some \(s\in {\mathbb {N}}\) . In this case we can assume that \(\mathcal {X}=\mathcal {Z}_s(T)\) ,The assumption that the sequences \(a_1,\ldots , a_\ell \)  are very good for seminorm estimates (versus simply “good”) is needed in order to get this reduction. hence using Theorem REF  and an approximation argument we can assume that the system \((X,\mu ,T)\)  has finite rational spectrum, in which case there exists \(k\in {\mathbb {N}}\)  such the system \((X,\mu ,T^k)\)  has trivial rational spectrum.
For \(i=1,\ldots , \ell \)  we let
\(b_i(n):=\sum _{r=0}^{k-1} {\bf 1}_{k{\mathbb {Z}}+r}(a_i(n)) \, \frac{a_i(n)-r}{k}, \quad n\in {\mathbb {Z}}.\) 
By Lemma REF , the sequences \(b_1,\ldots , b_\ell \)  are good for irrational equidistribution and seminorm estimates.
Then for \(i=1,\ldots , \ell \)  and \(S:=T^k\)  we have that
\(T^{a_i(n)}f_i=\sum _{r=0}^{k-1} {\bf 1}_{k{\mathbb {Z}}+r}(a_i(n))\, (T^k)^{\frac{a_i(n)-r}{k}}(T^rf_i)=\sum _{r=0}^{k-1} {\bf 1}_{k{\mathbb {Z}}+r}(a_i(n))\, S^{b_i(n)}(T^rf_i), \quad n\in {\mathbb {N}}.\) 
We insert this identity in (REF ) and expand the product.
We deduce
that it suffices to show the following: If the (not necessarily ergodic) system \((X,\mu ,S)\)  has trivial rational spectrum, the sequences \(b_1,\ldots , b_\ell \colon {\mathbb {N}}\rightarrow {\mathbb {Z}}\)  are good for irrational equidistibution and seminorm estimates,
and \({\mathbb {E}}(f_1|\mathcal {K}(T))=0\) , then for every bounded sequence \((c_n)\)  we have
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, c_n\cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell =0\) 
in \(L^2(\mu )\) .
To prove this, we first remark that since the system \((X,\mu ,S)\)  has trivial rational spectrum, the same holds for the system
\((X\times X,\mu \times \mu ,S\times S)\) . Furthermore, it is known that
\({\mathbb {E}}(f_1|\mathcal {K}(S))=0\)  implies that
\({\mathbb {E}}(f_1\otimes \overline{f}_1|\mathcal {I}(S\times S))=0\) . Combining these facts (whose proof follows for example from [17]}) we deduce that if
\(\mu \times \mu =\int (\mu \times \mu )_{(x,y)}\, d(\mu \times \mu )\) 
is the ergodic decomposition of the measure \(\mu \times \mu \)  with respect to the transformation \(S\times S\) , then
for \((\mu \times \mu )\) -almost every \((x,y)\in X\times X\)  the system \((X\times X,(\mu \times \mu )_{(x,y)},S\times S)\)  is ergodic, has trivial rational spectrum, and \(\int f_1\otimes \overline{f}_1\, d(\mu \times \mu )_{(x,y)}=0\) . By
Theorem REF  we deduce that for \((\mu \times \mu )\) -almost every \((x,y)\in X\times X\)  we have
\({\mathbb {E}}_{n\in [N]}\, (S\times S)^{b_1(n)}(f_1\otimes \overline{f}_1) \cdot \ldots \cdot (S\times S)^{b_\ell (n)}(f_\ell \otimes \overline{f}_\ell )\rightarrow ^{L^2((\mu \times \mu )_{(x,y)})}0\) 
as \(N\rightarrow \infty \) .
This implies that
\({\mathbb {E}}_{n\in [N]}\, (S\times S)^{b_1(n)}(f_1\otimes \overline{f}_1) \cdot \ldots \cdot (S\times S)^{b_\ell (n)}(f_\ell \otimes \overline{f}_\ell )\rightarrow ^{L^2(\mu \times \mu )}0\) 
as \(N\rightarrow \infty \) ,
and using the Cauchy-Schwarz inequality we deduce that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, \int (f_{0,N}\otimes \overline{f}_{0,N})\cdot (S\times S)^{b_1(n)}(f_1\otimes \overline{f}_1) \cdot \ldots \cdot (S\times S)^{b_\ell (n)}(f_\ell \otimes \overline{f}_\ell )\, d(\mu \times \mu )=0\) 
for all functions \(f_{0,N}\in L^\infty (\mu )\) , \(N\in {\mathbb {N}}\) , that are uniformly bounded.
Hence,
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, \Big |\int f_{0,N} \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell \, d\mu \Big |^2= 0,\) 
which implies using the Cauchy-Schwarz inequality that
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\,c_n\, \int f_{0,N} \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell \, d\mu = 0\) 
for every bounded sequence \((c_n)\) .
If we let
\(f_{0,N}:= \overline{{\mathbb {E}}_{n\in [N]}\, c_n \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell }, \quad N\in {\mathbb {N}},\) 
we deduce that
\(\lim _{N\rightarrow \infty }{\mathbb {E}}_{n\in [N]}\, c_n \cdot S^{b_1(n)}f_1 \cdot \ldots \cdot S^{b_\ell (n)}f_\ell = 0\) 
in \(L^2(\mu )\) . This completes the proof.

Proof of Corollary  REF
Let \(A\in \mathcal {X}\)  and \(\varepsilon >0\) . Let \(\mathcal {K}_r\)  denote the closed subspace of \(L^2(\mu )\)  consisting of all \(T^r\) -invariant functions. Then there exists \(r\in {\mathbb {N}}\)  such that
\(\left\Vert {\mathbb {E}}({\bf 1}_A|\mathcal {K}_{rat})-{\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r})\right\Vert _{L^2(\mu )}\le \frac{\varepsilon }{\ell }.\) 
Let
\(S_r=\lbrace n\in {\mathbb {N}}\colon a_1(n)\equiv 0, \ldots , a_\ell (n)\equiv 0 \! \! \! \pmod {r}\rbrace .\) 
By assumption, we have \(\bar{d}(S_r)>0\) , hence there exist \(N_k\rightarrow \infty \)  such that
\(\lim _{k\rightarrow \infty }\frac{|S_r\cap [N_k]|}{N_k}>0.\) 
First, we claim that \(\mathcal {K}_{rat}(T)\)  is a characteristic factor for the averages
\({\mathbb {E}}_{n\in S_r\cap [N_k]}\, T^{a_1(n)}f_1\cdot \ldots \cdot T^{a_\ell (n)}f_\ell ,\) 
meaning, if \({\mathbb {E}}(f_{j_0}|\mathcal {K}_{rat}(T))=0\)  for some \(j_0\in \lbrace 1,\ldots , \ell \rbrace \) , then the averages converge to zero in \(L^2(\mu )\)  as \(k\rightarrow \infty \) .
To see this, note that for every \(k\in {\mathbb {N}}\)  the averages (REF ) are equal to
\(\frac{N_k}{|S_r\cap [N_k]|}\cdot {\mathbb {E}}_{n\in [N_k]}\, (T\times R)^{a_1(n)}(f_1\otimes g)\cdot \ldots \cdot T^{a_\ell (n)}(f_\ell \otimes g),\) 
where \(R\)  is the shift transformation on a cyclic group of order \(r\)  and \(g\)  is the indicator function of the identity element in this cyclic group.
Then \({\mathbb {E}}(f_{j_0}\otimes g|\mathcal {K}_{rat}(T\times R))=0\) , and since by assumption the sequences \(a_1,\ldots , a_\ell \)  are very good for seminorm estimates and
good for irrational equidistribution, we get by Theorem REF  that the averages in (REF ) converge to 0 in \(L^2(\mu )\)  as \(k\rightarrow \infty \) .
We deduce from this and (REF ) that the averages (REF ) converge to 0 in \(L^2(\mu )\)  as \(k\rightarrow \infty \) , completing the proof of our claim.
It follows from what we just proved that the limit
\(\liminf _{k\rightarrow \infty }{\mathbb {E}}_{n\in S_r\cap [N_k]}\, \mu (A\cap T^{-a_1(n)}A\cap \cdots \cap T^{-a_\ell (n)}A)\) 
is equal to the limit
\(\liminf _{k\rightarrow \infty }{\mathbb {E}}_{n\in S_r\cap [N_k]}\int {\mathbb {E}}({\bf 1}_A|\mathcal {K}_{rat}(T))\cdot T^{a_1(n)}{\mathbb {E}}({\bf 1}_A|\mathcal {K}_{rat}(T))\cdot \ldots \cdot T^{a_\ell (n)}{\mathbb {E}}({\bf 1}_A|\mathcal {K}_{rat}(T))\, d\mu .\) 
Using (REF ) and telescoping, we get that the last limit is greater or equal than
\(\liminf _{k\rightarrow \infty }{\mathbb {E}}_{n\in S_r\cap [N_k]}\int {\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r}(T))\cdot T^{a_1(n)}{\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r}(T))\cdot \ldots \cdot T^{a_\ell (n)}{\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r}(T))\, d\mu -\varepsilon .\) 
Note that for \(n\in S_r\)  we have \(T^{a_j(n)}{\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r}(T))={\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r}(T))\) , \(j=1,\ldots , \ell \) , hence the last limit is equal to
\(\int ({\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r}(T)))^{\ell +1}\, d\mu \ge \Big (\int {\mathbb {E}}({\bf 1}_A|\mathcal {K}_{r}(T))\, d\mu \Big )^{\ell +1}=(\mu (A))^{\ell +1}.\) 
Combining the above estimates we get that
\(\liminf _{k\rightarrow \infty }{\mathbb {E}}_{n\in S_r\cap [N_k]}\, \mu (A\cap T^{-a_1(n)}A\cap \cdots \cap T^{-a_\ell (n)}A)\ge (\mu (A))^{\ell +1}-\varepsilon ,\) 
completing the proof of Corollary REF .

Joint ergodicity of special sequences
In this section we prove the results of Sections REF  and REF .
Definition of Hardy fields
Let \(B\)  be the collection of equivalence classes of real valued
functions defined on some half line \([c,+\infty )\) , where we
identify two functions if they agree eventually.The
equivalence classes just defined are often called “germs of
functions”. We choose to use the word function when we refer to
elements of \(B\)  instead, with the understanding that all the
operations defined and statements made for elements of \(B\)  are
considered only for sufficiently large values of \(t\in {\mathbb {R}}\) .
A
Hardy field \(\mathcal {H}\)  is a subfield of the ring \((B,+,\cdot )\)  that is
closed under differentiation.
For the purposes of this section we assume that all Hardy
fields \(\mathcal {H}\)  considered are contained in some other Hardy field \(\mathcal {H}^{\prime }\)  that satisfies property (REF ).
A particular example of such a Hardy field \(\mathcal {H}\)  is the collection of
logarithmic-exponential functions, meaning all functions
defined on some half line \([c,+\infty )\)  by a finite combination of
the symbols \(+,-,\times , :, \log , \exp \) , operating on the real
variable \(t\)  and on real constants; linear combinations of functions of the form \(t^a(\log {t})^b c^t\) , \(a,b\in {\mathbb {R}}\) , \(c>0\) , are examples of such functions.
The reader can find more information about them in [18]} and the references therein.

Good equidistribution properties for Hardy field sequences
We will use the following equidistribution result:
Theorem 6.1 (Boshernitzan [18]}) 
Let \(a\colon [c,+\infty )\rightarrow {\mathbb {R}}\)  be a Hardy field function with at most polynomial growth. Then the sequence
\((a(n))\)  is equidistributed on \( if and only if it stays logarithmically away from rational polynomials (see definition in Section~\ref {SS:16}).\) 
We will also use the following reduction, variants of which have been frequently used in the literature. We give its proof for completeness.
Lemma 6.2 
Let \(a_1,\ldots , a_\ell \colon [c,+\infty )\rightarrow {\mathbb {R}}\)  be such that for every
\(t_1,\ldots , t_\ell \in {\mathbb {R}}\) , not all of them zero, we have
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, e(a_1(n)t_1+\cdots + a_\ell (n)t_\ell ) =0.\) 
Then the sequences \([a_1(n)],\ldots , [a_\ell (n)]\)  are good for equidistribution.

Remark If we assume that (REF ) holds for all \(t_1,\ldots , t_\ell \in {\mathbb {R}}\) , not all of them
rational, then a similar argument gives that the sequences \([a_1(n)],\ldots , [a_\ell (n)]\)  are good for irrational equidistribution.

We first remark that it suffices to show the following: If \(F\colon  and \) GR\( are Riemann-integrable, thenfor all \) t1,..., t[0,1)\(, not all of them \) 0\(, we have\begin{equation}\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, F(a_1(n)t_1+\cdots + a_\ell (n)t_\ell ) \, G(a_1(n),\ldots , a_\ell (n))=\int F\, dm_\int G \, dm_{\ell }.\end{equation}Indeed, if this is the case, then using (\ref {E:wantedb})for the continuous function \) F(x):=e(x)\(, \) x, and the Riemann integrable function \(G(x_1,\ldots , x_\ell ):=e(-\lbrace x_1\rbrace t_1-\cdots - \lbrace x_\ell \rbrace t_\ell )\) , \(x_1,\ldots , x_\ell \in ,we get that\)\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\, e([a_1(n)]t_1+\cdots + [a_\ell (n)]t_\ell )=0\)\(for all \) t1,..., t[0,1)\(, not all of them \) 0\(.\) We move now to the proof of (). After approximating from above and below by continuous functions, we can assume that \(F\in C(\)  and \(G\in C(\ell )\) . After a further approximation by trigonometric polynomials we can assume that \(F(x)=e(kx)\) , \(x\in , and \) G(x1,..., x)=e(k1x1++ kx)\(, \) x1,..., x, for some \(k,k_1,\ldots , k_\ell \in {\mathbb {Z}}\) .
If \(k=k_1=\cdots =k_\ell =0\) , then the identity is obvious. Hence, it suffices to show that for all \(k,k_1,\ldots , k_\ell \in {\mathbb {Z}}\) , not all of them zero, and \(t_1,\ldots , t_\ell \in [0,1)\) , not all of them zero, we have
\(\lim _{N\rightarrow \infty } {\mathbb {E}}_{n\in [N]}\,e((k_1+kt_1)a_1(n)+\cdots + (k_\ell +kt_\ell ) a_\ell (n))=0.\) 
Since \(k_1+kt_1,\ldots , k_\ell +kt_\ell \)  are not all of them zero, this follows from our assumptions, completing the proof.
Combining the previous two results we get the following:
Proposition 6.3 
Let \(a_1,\ldots , a_\ell \colon [c,+\infty )\rightarrow {\mathbb {R}}\)  be functions of at most polynomial growth from a Hardy field such that every non-trivial linear combination of these functions stays logarithmically away from rational polynomials.
Then
the sequences \([a_1(n)],\ldots , [a_\ell (n)]\)  are good for equidistribution.

In a similar fashion, using the variant recorded on the remark following Lemma REF ,
we get the following:
Proposition 6.4 
Let \(a_1,\ldots , a_\ell \colon [c,+\infty )\rightarrow {\mathbb {R}}\)  be functions of at most polynomial growth from a Hardy field such that every non-trivial linear combination of these functions, with at least one irrational coefficient,
stays logarithmically away from rational polynomials.
Then
the sequences \([a_1(n)],\ldots , [a_\ell (n)]\)  are good for irrational equidistribution.

Good seminorm estimates for Hardy field sequences
We will use the following known result.As far as we know this result is proved in detail only under the assumption (REF ), which is the reason why we impose this assumption on all Hardy fields considered in this section.
Proposition 6.5 
Let \(a_1,\ldots , a_\ell \colon [c,+\infty )\rightarrow {\mathbb {R}}\)  be functions from a Hardy field such that
the functions and their pairwise differences are non-constant functions in \(\mathcal {T}+\mathcal {P}\) . Then the sequences
\([a_1(n)],\ldots , [a_\ell (n)]\)  are very good for seminorm estimates.

If \(\mathcal {T}\)  is replaced with the class of functions \(a\colon {\mathbb {R}}_+\rightarrow {\mathbb {R}}\)  that satisfy the slightly more restrictive growth condition \(t^{k+\varepsilon }\prec a(t)\prec t^{k+1}\)  for some \(k\in {\mathbb {Z}}_+\) , then the argument used to prove [20]} can be applied without any change to prove Proposition REF . For the more extended class of functions used above one can employ the argument used to prove [21]} without essential changes. We omit the details.

Proof of Theorem  REF
By Proposition REF  the sequences \([a_1(n)],\ldots , [a_\ell (n)]\)  are good for seminorm estimates, and by Proposition REF  they are also good for equidistribution. Hence by Theorem REF  they are jointly ergodic.

Proof of Theorem  REF
By Proposition REF  the sequences \([a_1(n)],\ldots , [a_\ell (n)]\)  are very good for seminorm estimates and by Proposition REF  they are good for irrational equidistribution. Hence, by Theorem REF 
they are jointly ergodic for totally ergodic systems and by Theorem REF  the rational Kronecker factor is characteristic for these sequences.

Proof of Theorem  REF
By Corollary REF  it suffices to show that these collections of sequences are good for equidistribution. By Lemma REF  it suffices to show that, after removing the integer parts, every non-trivial linear combination of the given collections of sequences is equidistributed on the circle.
For the sequences in Part \((i)\)  this follows by combining [22]} (which is the main result in [23]}) with [22]}.
For the sequences in Parts \((ii)\)  and \((iii)\)  this follows from
[25]} and [26]}.

Joint ergodicity for flows
In this section we prove Theorem REF , which we repeat for convenience.
Theorem 7.1 Let \(a_1,\ldots , a_\ell \colon [c,+\infty )\rightarrow {\mathbb {R}}_+\)  be functions from a Hardy field. Suppose that there exists \(\delta >0\)  such that \(t^\delta \prec a_1(t)\)  and
\((a_{j+1}(t))^\delta \prec a_j(t)\prec (a_{j+1}(t))^{1-\delta }\)  for \(j=1,\ldots , \ell -1\) .
Then for all measure preserving actions \(T_1^t,\ldots , T_\ell ^t\) , \(t\in {\mathbb {R}}\) , on a probability space \((X,\mathcal {X},\mu )\)  and \(f_1,\ldots , f_\ell \in L^\infty (\mu )\) , we have
\(\lim _{y\rightarrow +\infty } \frac{1}{y} \int _0^y f_1(T_1^{a_1(t)}x)\cdot \ldots \cdot f_\ell (T_\ell ^{a_{\ell }(t)}x)\, dt=\tilde{f_1}\cdots \tilde{f_\ell }\) 
pointwise for \(\mu \) -almost every \(x\in X\) , where for \(j=1,\ldots , \ell \)  we denote by \(\tilde{f_j}\)  the orthogonal projection of \(f_j\)  on the space of functions that are \(T_j^t\) -invariant for every \(t\in {\mathbb {R}}\) .

We start with the following crucial change of variables property (a variant of this property also appears in [27]}):
Lemma 7.2 
Let \(a\colon {\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\)  be a function from a Hardy field such that \(t^\delta \prec a(t)\)  for some \(\delta >0\) .
Let
\(f\in L^\infty (m_{\mathbb {R}})\)  and suppose that the following limit exists
\(\lim _{y\rightarrow +\infty } \frac{1}{y}\int _0^y f(t)\, dt.\) 
Then also the following limit exists
\(\lim _{y\rightarrow +\infty } \frac{1}{y}\int _0^{y} f(a(t)) \, dt\) 
and the two limits are equal.

Remark More generally, our argument works if there exist \(\delta , M>0\)  such that the functions \(a,a^{-1}\)  are three times differentiable, their derivatives are non-zero on \([M,+\infty )\) , and also \(a(t)\ge t^\delta \)  and \(\frac{a(t)}{ta^{\prime }(t)}\)  is bounded for \(t\ge M\) .

We can assume that the first limit is 0.
Let
\(F(t)=\frac{1}{t}\int _0^t f(s)\, ds, \quad t>0.\) 
Let \(\varepsilon >0\) . Using our assumption we have that there exists \(M>0\)  such that \(|F(t)|\le \varepsilon \)  for \(t>M\)  and also \(a,a^{-1}\in C^3([M,+\infty ))\) .
Since \(f\)  is bounded, it suffices to show that \(\lim _{y\rightarrow +\infty }\frac{1}{y}\int _{a^{-1}(M)} ^y f(a(t))\, dt=0\) . We assume that \(y\)  is large enough so that \(a(y)\ge M\) . Using the change of variables \(s=a(t)\)  we get
\(\frac{1}{y}\int _{a^{-1}(M)} ^{y} f(a(t)) \, dt= \frac{1}{y}\int _M^{a(y)} f(t) \cdot (a^{-1})^{\prime }(t)\, dt.\) 
Since
\(f(t)=(t F(t))^{\prime }\) 
for Lebesgue almost every \(t\in {\mathbb {R}}\) ,
we have
\(\frac{1}{y}\int _M^{a(y)} f(t) \cdot (a^{-1})^{\prime }(t)\, dt= \frac{1}{y}\int _M^{a(y)} (t F(t))^{\prime } \cdot (a^{-1})^{\prime }(t)\, dt.\) 
Integration by parts (\(tF(t)\)  and \((a^{-1})^{\prime }(t)\)  are absolutely continuous on \([M,a(y)]\) ) gives that the last integral is equal to
\(\frac{a(y) F(a(y)) (a^{-1})^{\prime }(a(y))}{y}- \frac{C}{y}- \frac{1}{y}\, \int _M^{a(y)} t F(t) \cdot (a^{-1})^{\prime \prime }(t)\, dt\) 
for some \(C\in {\mathbb {R}}_+\) .
The first term in (REF ) is equal to
\(\frac{a(y) F(a(y)) (a^{-1})^{\prime }(a(y))}{y}=\frac{\int _0^{a(y)}f(t)\, dt}{ya^{\prime }(y)} =\frac{\int _0^{a(y)}f(t)\, dt}{a(y)}\cdot \frac{a(y)}{ya^{\prime }(y)}.\) 
Notice that since \(a(t)\)  is a Hardy field function with \(a(t)\succ t^\delta \) , we have that
\(\lim _{y\rightarrow +\infty } \frac{a(y)}{ya^{\prime }(y)}=\lim _{y\rightarrow +\infty }\frac{\log {y}}{\log {a(y)}}<\frac{1}{\delta }.\) 
Moreover, by assumption we have
\(\lim _{y\rightarrow +\infty } \frac{1}{y}\int _0^y f(t)\, dt=0\)  and \(\lim _{y\rightarrow +\infty }a(y)= +\infty \) . We deduce that the first term in (REF ) converges to 0 as \(y\rightarrow +\infty \) .
It remains to show that the limsup as \(y\rightarrow +\infty \)  of last term in (REF ) is bounded by a constant multiple of \(\varepsilon \) . Since \(|F(t)|\le \varepsilon \)  for \(t>M\) , it suffices to show that the limsup as \(y\rightarrow +\infty \)  of the expression
\(\frac{1}{y}\int _M^{a(y)} t \cdot |(a^{-1})^{\prime \prime }(t)|\, dt\) 
is bounded by a quantity that is independent of \(\varepsilon \)  . Since \((a^{-1})^{\prime \prime } \)  has eventually constant sign, say in \([M_1,+\infty )\) , we can replace \(M\)  with \(M_1\)  and remove the absolute value.
After doing so, integration by parts leads to the expression
\(\frac{1}{y}\Big ( a(y) \cdot (a^{-1})^{\prime }(a(y))-C_1 -\int _{M_1}^{a(y)} (a^{-1})^{\prime }(t)\, dt\Big )\) 
for some \(C_1\in {\mathbb {R}}_+\) .
The last expression is equal to
\(\frac{a(y)}{ya^{\prime }(y)} - \frac{C_1}{y} -\frac{y-a^{-1}(M_1)}{y}.\) 
As we showed before, the limit of the first term is bounded and the limit of the other two terms as \(y\rightarrow +\infty \)  is 1. This completes the proof.
We review some basic facts from the spectral theory of unitary \({\mathbb {R}}\) -actions that we will use. Proofs of the stated facts can be found for example in [28]}
(we use Theorem 9.58 and the variant of Theorems 9.17 that applies to flows). If \((X,\mu ,T^t)\) , \(t\in {\mathbb {R}}_+\) , is a measure preserving flow, then for every \(f\in L^2(\mu )\)  there exists a positive and bounded measure \(\sigma _f\)  on \({\mathbb {R}}\) , that is called the spectral measure of \(f\), such that
\(\int T^tf\cdot \overline{f}\, d\mu =\int e(ts)\, d\sigma _f(s), \quad t\in {\mathbb {R}}.\) 
More generally, for every \(f,g\in L^2(\mu )\)  there exists a complex measure \(\sigma _{f,g}\) , with bounded variation, such that
\(\int T^tf\cdot \overline{g}\, d\mu =\int e(ts)\, d\sigma _{f,g}(s), \quad t\in {\mathbb {R}}.\) 
Furthermore, if \(h\in L^\infty ({\mathbb {R}})\) , then there exists a bounded operator \(h(T)\colon L^2(\mu )\rightarrow L^2(\mu )\)  that commutes with \(T^t\) , \(t\in {\mathbb {R}}\) , and satisfies
\(\int h(T)f \cdot \overline{g}\, d\mu =\int h\, d\sigma _{f,g}\) 
for all \(f,g\in L^2(\mu )\) . We then have
\((h_1+h_2)(T)=h_1(T)+h_2(T), \quad h_1,h_2\in L^\infty (\mu ),\) 
and
\(d\sigma _{h(T)f}=h\, d\sigma _f, \quad \left\Vert h(T)f\right\Vert _{L^2(\mu )}=\left\Vert h\right\Vert _{L^2(\sigma _f)}.\) 
We will use the following fact:
Lemma 7.3 
Let \((X,\mu ,T^t)\) , \(t\in {\mathbb {R}}_+\) , be a measure preserving flow. Then the set
\(\mathcal {G}:=\lbrace f\in L^2(\mu )\colon \sigma _f \text{ has compact support}\rbrace \) 
is dense in \(L^2(\mu )\) .

Let \(f\in L^2(\mu )\) . Using the previous notation we have by (REF )
that for \(n\in {\mathbb {N}}\) 
the spectral measure of the function \(f_n={\bf 1 }_{[-n,n]}(T)f\)  is supported on the interval \([-n,n]\) , hence \(f_n\in \mathcal {G}\) . Moreover, since \({\bf 1}_{\mathbb {R}}(T)f=f\) , using (REF ) and (REF ) we get
\(\left\Vert f-f_n\right\Vert _{L^2(\mu )}=\left\Vert {\bf 1}_{{\mathbb {R}}\setminus [-n,n]}(T)f\right\Vert _{L^2(\mu )}=\left\Vert {{\bf 1}_{{\mathbb {R}}\setminus [-n,n]}}\right\Vert _{L^2(\sigma _f)}\rightarrow 0\) 
as \(n\rightarrow \infty \) , since \(\sigma _f\)  is a bounded measure.
Lemma 7.4 
Let \(b\colon {\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\)  be a function from a Hardy field that satisfies \(t^\delta \prec b(t)\prec t^{1-\delta }\)  for some \(\delta >0\) , and \((X,\mu ,T^t)\) , \(t\in {\mathbb {R}}_+\) , be a measure preserving flow. Then for every \(f\in L^\infty (\mu )\)  and \(c\in {\mathbb {R}}\)  we have
\(\lim _{y\rightarrow +\infty } \frac{1}{y}\int _0^y|f(T^{b(t+c)}x)-f(T^{b(t)}x)|^2\, dt=0\) 
for \(\mu \) -almost every \(x\in X\) .

Remark More generally, our argument works if the function \(b\)  satisfies the properties mentioned in the remark following Lemma REF  and also \(b(t)\prec t^{1-\delta }\)  and \(b^{\prime }\)  monotonically decreases to 0 as \(t\rightarrow +\infty \) .

We can assume that \(c\ge 0\) . Our assumptions imply that there exists \(t_0>0\)  such that for \(t\ge t_0\) 
we have \(0\le b^{\prime }(t)\le 1\) , \(b^{\prime }(t)\)  is decreasing, and \(b(t)\le t^{1-\delta }\) . Since \(f\in L^\infty (\mu )\)  it suffices to show that
\(\lim _{y\rightarrow +\infty } \frac{1}{y}\int _{t_0}^{y}|f(T^{b(t+c)}x)-f(T^{b(t)}x)|^2\, dt=0\) 
for \(\mu \) -almost every \(x\in X\) .
Let \(\varepsilon >0\) . If \(\mathcal {G}\)  is the dense subset of \(L^2(\mu )\)  given by Lemma REF ,
we have a decomposition
\(f=f_1+f_2\) 
where \(f_1\in \mathcal {G}\)  and \(\left\Vert f_2\right\Vert _{L^2(\mu )}\le \varepsilon \) .
For \(g\in L^2(\mu )\)  let
\(A_y(g):=\frac{1}{y}\int _{t_0}^y|g(T^{b(t+c)}x)-g(T^{b(t)}x)|^2\, dt, \quad y\ge t_0.\) 
We first deal with the contribution of \(f_2\) .
We clearly have
\(\limsup _{y\rightarrow +\infty } A_y(f_2) \le 4 \lim _{y\rightarrow +\infty } \frac{1}{y}\int _{t_0}^y|f_2(T^{b(t)}x)|^2\, dt.\) 
Hence, using Lemma REF  and the pointwise ergodic theorem for flows we deduce that
\(\int \limsup _{y\rightarrow +\infty } A_y(f_2) \, d\mu \le 4 \int \Big ( \lim _{y\rightarrow +\infty } \frac{1}{y}\int _{t_0}^y|f_2(T^tx)|^2\, dt\Big ) d\mu = 4\int |f_2|^2\, d\mu \le 4\varepsilon ^2.\) 
Next, we deal with the contribution of \(f_1\) .
Since \(f_1\in \mathcal {G}\) , there exists \(M>0\)  such that the spectral measure \(\sigma _{f_1}\)  of \(f_1\)  is supported on the set \([-M,M]\) .
Using the Fubini-Tonelli theorem we get
\(\left\Vert A_y(f_1)\right\Vert _{L^1(\mu )}=\frac{1}{y}\int _{t_0}^y\int _X|f_1(T^{b(t+c)}x)-f_1(T^{b(t)}x)|^2\,d\mu \, dt.\) 
Using (REF ) and the Fubini-Tonelli theorem again, we get that the last expression is equal to
\(\int _{-M}^M\frac{1}{y}\int _{t_0}^y|e(sb(t+c))-e(sb(t))|^2\, dt\, d\sigma _{f_1}(s).\) 
After bounding the integrant pointwise, using the mean value theorem, the fact that \(b^{\prime }\)  is non-negative, and decreasing for \(t\ge t_0\) , and that \(c\ge 0\) , we get that the last expression is bounded by a constant multiple of
\(\int _{-M}^M\frac{1}{y}\int _{t_0}^y \big (scb^{\prime }(t))^2\, dt\, d\sigma _{f_1}(s)\le M^2c^2\left\Vert f_1\right\Vert ^2_{L^2(\mu )}\frac{1}{y}\int _{t_0}^y (b^{\prime }(t))^2\, dt.\) 
Since \(0\le b^{\prime }(t)\le 1\)  for \(t\ge t_0\)  we have for \(y\ge t_0\)  that
\( \int _{t_0}^y \big (b^{\prime }(t))^2dt\le \int _{t_0}^y b^{\prime }(t)\, dt =b(y)-b(t_0).\) 
Since \(b(y)\le y^{1-\delta }\)  for \(y>t_0\) , combining the above we get for \(C:=Mc^2\vert \!|\!| f_2|\!|\!\vert ^2_{L^2(\mu )}\)  that
\(\left\Vert A_y(f_1)(x)\right\Vert _{L^1(\mu )}\le \frac{C}{y^\delta }\) 
for all \(y>t_0\) . Using the Borel-Cantelli lemma, we get that for \(a>1/\delta \)  we have \(\lim _{N\rightarrow \infty } A_{N^a}(f_1)(x)= 0\)  for \(\mu \) -almost every \(x\in X\) .
From this we deduce that
\(\lim _{y\rightarrow +\infty } A_y(f_1)(x)=0\) 
for \(\mu \) -almost every \(x\in X\) . Indeed, if \(N_y\in {\mathbb {Z}}_+\)  is such that \(N_y^a\le y<(N_y+1)^a\) , then
\(\left\Vert A_y(f_1)-A_{N_y^a}(f_1)\right\Vert _{L^\infty (\mu )}\le 2\, \left\Vert f_1\right\Vert _{L^\infty (\mu )} |1-N_y^a/y|\rightarrow 0\)  as \(y\rightarrow +\infty \) .
Combining (REF ) and (REF ) we get that
\(\int \limsup _{y\rightarrow +\infty } A_y(f) \, d\mu \le 8\varepsilon ^2.\) 
Since \(\varepsilon \)  is arbitrary, we get that (REF ) holds for \(\mu \) -almost every \(x\in X\) , completing the proof.
We prove the statement by induction on \(\ell \) . For \(\ell =1\)  the result follows from Lemma REF  and the pointwise ergodic theorem for flows. Suppose that the statement holds for \(\ell -1\) , we shall show that it holds for \(\ell \) .
Without loss of generality we can assume that
\(\left\Vert f_j\right\Vert _{L^\infty (\mu )}\le 1\)  for \(j=1,\ldots , \ell \) .
We let
\(A_y(f_\ell )(x):= \frac{1}{y} \int _0^y f_1(T_1^{a_1(t)}x)\cdot \ldots \cdot f_\ell (T_\ell ^{a_{\ell }(t)}x)\, dt, \quad y\in {\mathbb {R}}_+.\) 
Let \(\varepsilon >0\) . Using a standard Hilbert space argument we get a decomposition
\(f_\ell =f_{\ell ,1}+f_{\ell ,2}+f_{\ell ,3},\) 
where \(f_{\ell , 1}=\tilde{f_\ell }\) , \(\left\Vert f_{\ell ,2}\right\Vert _{L^2(\mu )}\le \varepsilon \) , and
\(f_{\ell ,3}\)  belongs to the linear subspace spanned by the functions \(T_\ell ^ch-h\) , \(c\in {\mathbb {R}}\) , \(h\in L^2(\mu )\) . After approximating \(h\)  in \(L^2(\mu )\)  by functions in \(L^\infty (\mu )\)  and incorporating the error in \(f_{\ell ,2}\) , we can assume that \(h\in L^\infty (\mu )\) . Furthermore, we have
\(\left\Vert f_{\ell , 1}\right\Vert _{L^\infty (\mu )}\le \left\Vert f_{\ell }\right\Vert _{L^\infty (\mu )}\le 1\)  and
\(\left\Vert f_{\ell ,3}\right\Vert _{L^2(\mu )}\le 2+\varepsilon \) . An application of the Cauchy-Schwarz inequality and the Fubini-Tonelli theorem shows that for \(\mu \) -almost every \(x\in X\)  the
quantities \(A_y(f_{\ell ,j})\) , \(j=1,2,3\) , are well defined finite numbers.
Our goal is to show (REF ) below. We first deal with the contribution of the term \(f_{\ell ,1}\) . Since \(T_\ell ^tf_{\ell , 1}=f_{\ell ,1}=\tilde{f_\ell }\)  for every \(t\in {\mathbb {R}}_+\) , the induction hypothesis gives
\(\lim _{y\rightarrow +\infty } A_y(f_{\ell ,1})=\tilde{f_1}\cdots \tilde{f_\ell }.\) 
for \(\mu \) -almost every \(x\in X\) .
Next we deal with the contribution of the term \(f_{\ell ,2}\) .
For every \(y>0\) , using the Cauchy-Schwarz inequality we have
\(|A_y(f_{\ell ,2})(x)|^2\le \frac{1}{y}\int _0^y |f_{\ell ,2}(T_\ell ^{a_\ell (t)}x)|^2\, dt.\) 
Using Lemma REF  and
the pointwise ergodic theorem for flows we get
\(\int \limsup _{y\rightarrow +\infty }|A_y(f_{\ell ,2})|\, d\mu =\left\Vert f_{\ell ,2}\right\Vert ^2_{L^2(\mu )} \le \varepsilon .\) 
It remains to deal with the contribution of the term \(f_{\ell ,3}\) . We claim that
\(\lim _{y\rightarrow +\infty }|A_y(f_{\ell ,3})(x)|=0\) 
for \(\mu \) -almost every \(x\in X\) .
By Lemma REF ,We remark that since not all Hardy fields are closed under composition and compositional inversion, we cannot assume that the functions \(b_j\)  belong to some Hardy field. Nevertheless, one can easily verify that these functions satisfy the necessary assumptions mentioned on the remark following Lemma REF , so we are entitled to apply this lemma. it suffices to show that
for \(\mu \) -almost every \(x\in X\)  we have
\(\lim _{y\rightarrow +\infty }\frac{1}{y}\int _0^{y} \prod _{j=1}^{\ell -1}f_j(T_j^{b_j(t)}x)\cdot f_{\ell ,3}(T_\ell ^{t}x)\, dt=0\) 
where \(b_j:=a_j\circ a^{-1}_\ell \)  for \(j=1,\ldots , \ell -1\) .
In order to establish (REF ) it suffices to verify that if
\(f_{\ell ,3}:=T_\ell ^ch- h,\) 
for some \(c\in {\mathbb {R}}\)  and \(h\in L^\infty (\mu )\) , then (REF ) holds for \(\mu \) -almost every \(x\in X\) .
So let \(c>0\)  and \(h\in L^\infty (\mu )\) .
After inserting \( f_{\ell ,3}=T_\ell ^ch- h\)  in (REF ) and using the change of variables \(t\mapsto t-c\)  in the first of the two integrals, we get that it suffices to show that
\(\lim _{y\rightarrow +\infty }\frac{1}{y}\int _0^{y} \Big (\prod _{j=1}^{\ell -1}f_j(T_j^{b_j(t-c)}x) - \prod _{j=1}^{\ell -1}f_j(T_j^{b_j(t)}x)\Big ) \cdot h(T_\ell ^{t}x)\, dt=0\) 
for \(\mu \) -almost every \(x\in X\) . It suffices to show that for
\(d=1,\ldots , \ell -1\)  we have
\(\lim _{y\rightarrow +\infty }\frac{1}{y}\int _0^{y} \Big (\prod _{j=1}^{d}f_j(T_j^{b_j(t-c)}x)\prod _{j=d+1}^{\ell -1}f_j(T_j^{b_j(t)}x) -\\ \prod _{j=1}^{d-1}f_j(T_j^{b_j(t-c)}x)\prod _{j=d}^{\ell -1}f_j(T_j^{b_j(t)}x)\Big ) \cdot h(T_\ell ^{t}x)\, dt=0\) 
for \(\mu \) -almost every \(x\in X\) .
Finally, notice that our growth assumptions give that \(t^{\delta ^{\prime }}\prec b_d(t)\prec t^{1-\delta ^{\prime }}\)  for some \(\delta ^{\prime }>0\) .
Hence, by Lemma REF ,Again here, in order to avoid the assumption that the function \(b_d\)  belongs to some Hardy field, we can verify that the assumptions mentioned on the remark following Lemma REF 
are satisfied. for \(d=1,\ldots , \ell -1\) , we have
\(\lim _{y\rightarrow +\infty } \frac{1}{y}\int _0^y|f_d(T_d^{b_d(t-c)}x)-f_d(T_d^{b_d(t)}x)|^2\, dt=0\) 
for \(\mu \) -almost every \(x\in X\) .
Hence, using the Cauchy-Schwarz inequality, equation (REF ), and the fact that all the functions \(f_j\)  and \(h\)  are bounded, we get that (REF ) holds for \(\mu \) -almost every \(x\in X\) . Combning the above we deduce that (REF ) holds.
From (REF ), (REF ), (REF ), we deduce that
\(\int \limsup _{y\rightarrow +\infty }|A_y(f_{\ell })-\tilde{f_1}\cdots \tilde{f_\ell }|\, d\mu \le \varepsilon .\) 
Since \(\varepsilon \)  was arbitrary, we deduce that
\(\lim _{y\rightarrow +\infty } A_y(f_{\ell })(x)=\tilde{f_1}\cdots \tilde{f_\ell }\) 
for \(\mu \) -almost every \(x\in X\) , completing the proof.
